Classification of non-Kähler surfaces and
locally conformally Kähler geometry
Liviu Ornea111Liviu Ornea and Victor Vuletescu are partially supported by a
grant of Ministry of Research and Innovation, CNCS - UEFISCDI, project number
PN-III-P4-ID-PCE-2016-0065, within PNCDI III., Misha Verbitsky222Misha
Verbitsky is partially supported by the Russian Academic Excellence Project
’5-100” and CNPq - Process 313608/2017-2.
Keywords: locally conformally Kähler, surfaces, Kato surface, elliptic
fibration. 2010 Mathematics Subject Classification: 53C55, 32E05, 32E10.
and Victor Vuletescu1.

Abstract

We give a mostly self-contained proof of the classification of non-Kähler
surfaces based on Buchdahl-Lamari theorem. We also prove that all non-Kähler
surfaces which are not of class VII are locally conformally Kähler.

###### Contents

  1. 1 Introduction
    1. 1.1 Buchdahl-Lamari theorem
    2. 1.2 Locally conformally Kähler surfaces
  2. 2 Cohomology of non-Kähler surfaces
    1. 2.1 Bott-Chern cohomology of a surface
    2. 2.2 First cohomology of non-Kähler surfaces
    3. 2.3 Second cohomology of non-Kähler surfaces
    4. 2.4 Vanishing in multiplication of holomorphic 1-forms
    5. 2.5 Structure of multiplication in $H^{1}(M)$ for non-Kähler surface without curves
  3. 3 Barlet spaces
  4. 4 Non-Kähler elliptic surfaces
    1. 4.1 Elliptic fibrations on non-Kähler surfaces
    2. 4.2 Isotrivial elliptic fibrations
    3. 4.3 Blanchard theorem and LCK structure on non-Kähler elliptic surfaces
  5. 5 Class VII surfaces
  6. 6 Brunella’s theorem: all Kato surfaces are LCK

##  1 Introduction

This paper appeared as a distillation of a lecture course on complex surfaces
given in 2008 and 2012 in Moscow Independent University. The main reference on
complex surfaces is the great book by Barth, Hulek, Peters and Van de Ven
[BHPV]. This book offers a powerful narrative, but (as it often happens with
great books) some plots are nested within more plots, and it sometimes becomes
hard to disjoin a particular strain from the polyphonic discourse.

For a forthcoming book on locally conformally Kähler (LCK) manifolds the first
named two authors needed a classification of LCK structures on (a posteriori,
non-Kähler) surfaces. It was easier (an more enlightening) to prove the non-
Kähler part of the Kodaira-Enriques classification directly along with the
classification of LCK structures.111For a definiton and an introduction to LCK
structures, see Subsection 1.2. There is (almost) no new results of this
paper, but most of the proofs are different from that given in the literature,
such as [BHPV].

We tried to keep this exposé self-contained. With the exception of two results
from Lamari’s paper [L], we invoke only general notions of complex algebraic
geometry, found, for example, in [D2].

The main focus of this paper is elliptic non-Kähler surfaces. We give a new
proof that they are principal elliptic bundles in the orbifold category (4.1)
- a result originally proven by Brînzănescu in [Bri1] ( see also [Bri2]) -,
and locally conformally Kähler (4.3) - a result due to Belgun, [Bel]. We prove
that all these surfaces are Vaisman, giving a new proof of Belgun’s
classification of Vaisman surfaces ([Bel]).

We prove that all non-Kähler non-elliptic surfaces are of class VII (5). There
is enough good literature ([DOT], [Dl], [Na1], [Na2], [T4]) on class VII
surfaces for us to give less attention to this case.

We state the known classification results up to the GSS conjecture and give a
new proof of Brunella’s theorem on existence of LCK metrics on Kato surfaces.
Together with the GSS conjecture (still not fully proven) this would imply
that all non-Kähler complex surfaces are LCK, with the exception of some of
the Inoue surfaces ([Bel]).

###  1.1 Buchdahl-Lamari theorem

In [Bu, L], N. Buchdahl and A. Lamari have proven a result previously known
only from the Kodaira-Enriques classification of complex surfaces.

Theorem 1.1:  Let $M$ be a compact complex surface. Then the first Betti
number $b_{1}(M)$ is odd if and only if $M$ is non-Kähler.

Its direct proof, however, simplifies this classification significantly. In
this paper we attempt to recover most of the Kodaira-Enriques classification
for non-Kähler surfaces using the Buchdahl-Lamari theorem and the following
intermediate result (1.1), which was used by Lamari to prove 1.1. Our proof is
different from the classical one, found in [BHPV], in a few aspects: we do not
rely on birational arguments and classification of the elliptic surfaces due
to Kodaira. Also, we aim to classify the locally conformally Kähler structures
on complex surfaces.

Theorem 1.2:  Let $M$ be a compact complex non-Kähler surface. Then there
exists a positive, exact (1,1)-current $\Theta$ on $M$.

Proof: [L, Theorem 6.1].

Remark 1.3:  The existence of an exact, positive (1,1)-current $\Theta$
immediately implies that the surface $M$ is non-Kähler. Indeed, suppose that
$M$ admits a Kähler form $\omega$. Then $\int_{M}\omega\wedge\Theta>0$, which
is impossible because $\Theta$ is exact.

###  1.2 Locally conformally Kähler surfaces

Let us recall that a complex manifold $M$, $\dim_{\mathbb{C}}M>1$, is called
locally conformally Kähler (LCK) if it admits a Hermitian form $\omega$ such
that $d\omega=\theta\wedge\omega$, where $\theta$ is a closed 1-form, see
[DO]. Then any cover
$\tilde{M}\stackrel{{\scriptstyle\tau}}{{{\>\longrightarrow\>}}}M$ such that
$\tau^{*}(\theta)=df$ is exact is Kähler, with the Kähler form given by
$\tilde{\omega}:=e^{-f}\tau^{*}\omega$. When $\theta$ is exact, $M$ is called
globally conformally Kähler. However, when $\theta$ is not exact, and $M$ is
compact, $M$ is non-Kähler ([Va1]). If, in addition, there exists a
holomorphic conformal flow
$\rho:\;{\mathbb{C}}{\>\longrightarrow\>}\operatorname{Aut}(M)$ acting on the
Kähler cover $(\tilde{M},\tilde{\omega})$ non-isometrically, $M$ is called
Vaisman.

Remark 1.4: Since $(\tilde{M},\tilde{\omega})$ is Kähler, any LCK manifold can
be obtained as a quotient of a Kähler manifold by a discrete, proper action of
a group of holomorphic automorphisms, acting on $\omega$ by holomorphic
homotheties.222Notice that any conformal holomorphic map of a connected Kähler
manifold
$\varphi:\;(\tilde{M},\tilde{\omega}){\>\longrightarrow\>}(\tilde{M},\tilde{\omega})$,
with $\dim_{\mathbb{C}}\tilde{M}>1$, takes $\tilde{\omega}$ to
$f\tilde{\omega}$, with $df=0$ because
$d\varphi^{*}\tilde{\omega}=d(f\tilde{\omega})=df\wedge\omega=0$. Therefore,
any conformal automorphism of a Kähler manifold is a homothety. Thus, an LCK
manifold can be defined as a quotient of a Kähler manifold
$(\tilde{M},\tilde{\omega})$ by a discrete, proper group acting by holomorphic
homotheties.

Definition 1.5: Let $M$ be a compact complex surface with $b_{1}(M)=1$. It is
called a class VII surface if its Kodaira dimension is $\kappa(M)=-\infty$.

From the Kodaira-Enriques classification ([BHPV]) it follows that all minimal
non-Kähler surfaces not of class VII are elliptic. We prove this result
(independently from the rest of Kodaira-Enriques classification) in 4.1. We
prove that all elliptic surfaces are Vaisman when they are minimal.

Theorem 1.6: Let $M$ be a compact, non-Kähler surface, which is not of class
VII. Then $M$ admits an LCK structure, and a Vaisman one if $M$ is minimal.

Proof: By 5, $M$ is elliptic, when it is minimal, and by 4.3, it is Vaisman.
By [Tr] and [Vu1], the blow-up at points preserves the LCK class, in
particular a blow-up of an LCK surface remains LCK. Therefore, a surface is
LCK if its minimal model is LCK, for example elliptic.

For class VII surfaces, a complete classification is not known, but it would
follow from the so-called “Global Spherical Shell conjecture” (GSS
conjecture), which claims that any minimal class VII surface $M$ with
$b_{2}>0$ contains an open complex subvariety $U\subset M$ biholomorphic to a
neighbourhood of the standard sphere $S^{3}\subset{\mathbb{C}}^{2}$ such that
$M\backslash U$ is connected. Surfaces which satisfy this conjecture are
called Kato surfaces.

Brunella has shown that all Kato surfaces are LCK ([Bru2] and Section 6).
Bogomolov’s theorem on class VII surfaces with $b_{1}(M)=0$ (see [Bo1, Bo2,
LYZ, T1]) implies that they are either Inoue surfaces or Hopf surfaces. The
Hopf surfaces are LCK ([GO, Bel, OV2]), and among the three classes of Inoue
surfaces, two are LCK, and the third contains a subclass which does not admit
an LCK structure ([Tr, Bel]).

The modern proof of Bogomolov’s classification theorem (due to A. Teleman and
Li-Yau-Zhang) is based on gauge theory. Using gauge-theoretic methods, A.
Teleman was able to prove the GSS conjecture for minimal class VII surfaces
with $b_{2}=1$ ([T3]). Extending this approach to $b_{1}>1$, Teleman was also
able to prove that any class VII manifold with $b_{1}(M)=2$ contains a cycle
of rational curves, hence can be smoothly deformed to a blown-up Hopf surface
([T4]).

Once the GSS conjecture is proven, this finishes the classification of LCK
surfaces. If it is true, all non-Kähler surfaces are LCK, except that
particular case of Inoue surfaces.

##  2 Cohomology of non-Kähler surfaces

###  2.1 Bott-Chern cohomology of a surface

In this section, we repeat some of the arguments about the Bott-Chern
cohomology previously given in [T2] and [ATV]. Recall that any compact complex
manifold admits a Gauduchon metric in any conformal class of Hermitian metrics
([Ga]). By definition, a Gauduchon metric on an $n$-dimensional complex
manifold is a Hermitian metric with Hermitian form $\omega$ satisfying
$dd^{c}(\omega^{n-1})=0$, where $d^{c}=IdI^{-1}$ is the twisted differential.

Definition 2.1: The Bott-Chern cohomology group of a complex manifold is
$H^{p,q}(M):=\displaystyle\frac{\ker
d{\left|{}_{{\phantom{|}\\!\\!}_{\Lambda^{p,q}(M)}}\right.}}{\operatorname{im}dd^{c}}$.

Remark 2.2: By the $dd^{c}$-lemma, on a compact Kähler manifold the Bott-Chern
cohomology groups are equal to de Rham cohomology groups. It is also easy to
see that the complex

$\Lambda^{p-1,q-1}(M)\stackrel{{\scriptstyle
dd^{c}}}{{{\>\longrightarrow\>}}}\Lambda^{p,q}(M)\stackrel{{\scriptstyle
d}}{{{\>\longrightarrow\>}}}\Lambda^{p+q+1}(M)$

is elliptic, hence the Bott-Chern cohomology is finite-dimensional on any
compact manifold.

Theorem 2.3:  Let $M$ be a compact non-Kähler surface. Then the kernel of the
natural map $P:\;H^{1,1}_{BC}(M){\>\longrightarrow\>}H^{2}(M)$ is
1-dimensional.

Proof. Step 1: Let $\omega$ be a Gauduchon metric on $M$. Consider the
differential operator $D:\;f\mapsto dd^{c}f\wedge\omega$ mapping functions to
4-forms. Clearly, $D$ is elliptic and its index is the same as the index of
the Laplacian: $\operatorname{\text{\sf ind}}D=\operatorname{\text{\sf
ind}}\Delta=0$, hence $\dim\ker D=\dim\operatorname{coker}D$. The Hopf maximum
principle implies that $\ker D$ only contains constants, hence
$\operatorname{coker}D$ is 1-dimensional. However,
$\int_{M}D(\alpha)=\int_{M}dd^{c}f\wedge\omega=\int_{M}fdd^{c}\omega=0$. This
implies that a 4-form $\kappa$ belongs to $\operatorname{im}D$ if and only if
$\int_{M}\kappa=0$.

Step 2: Let $\alpha$ be a closed $(1,1)$-form. Define
$\deg_{\omega}\alpha:=\int_{M}\omega\wedge\alpha$. Since
$\int_{M}dd^{c}f\wedge\omega=0$, this defines a map
$\deg_{\omega}:\;H^{1,1}_{BC}(M,{\mathbb{R}}){\>\longrightarrow\>}{\mathbb{R}}$.
Given a closed $(1,1)$-form $\alpha$ of degree 0, the form
$\alpha^{\prime}:=\alpha-dd^{c}(D^{-1}(\alpha\wedge\omega))$ satisfies
$\alpha^{\prime}\wedge\omega=0$, in other words, it is an $\omega$-primitive
(1,1)-form. For such forms, one has
$\alpha^{\prime}\wedge\alpha^{\prime}=-|\alpha^{\prime}|^{2}\omega\wedge\omega$,
giving

$\int_{M}\alpha^{\prime}\wedge\alpha^{\prime}=-\|\alpha^{\prime}\|^{2}_{\omega}$
(2.1)

which is impossible when $\alpha$ is a non-zero vector in $\ker P$. Therefore,
any two vectors in $\ker P$ are proportional.

###  2.2 First cohomology of non-Kähler surfaces

Lemma 2.4:  All holomorphic 1-forms on a compact complex surface are closed.

Proof: Let $\alpha\in\Lambda^{1,0}(M)$ be a holomorphic 1-form. Then
$\overline{\partial}\alpha=0$, because it is holomorphic, and by the same
reason $d\alpha$ is a holomorphic, exact (2,0)-form. Then $d\alpha\wedge
d\overline{\alpha}$ is a positive (2,2)-form, giving $0=\int_{M}d\alpha\wedge
d\overline{\alpha}=\|d\alpha\|^{2}$. Then $d\alpha=0$, and $\alpha$ is closed.

Theorem 2.5:  Let $M$ be a non-Kähler manifold and $\Theta$ an exact positive
(1,1)-current (1.1). Let $d\alpha=\beta$ be a (1,1)-form, $\alpha^{0,1}$ the
(0,1)-part of $\alpha$, and $A:=[\alpha^{0,1}]$ its Dolbeault class. Then:

(i)

Denote the space of holomorphic 1-forms on $M$ by $H^{1,0}(M)$, and let
$\overline{H^{1,0}(M)}$ denote its complex conjugate (the space of
antiholomorphic forms). By 2.2, holomorphic 1-forms are closed, therefore,
there is a natural map
$H^{1,0}(M)\oplus\overline{H^{1,0}(M)}{\>\longrightarrow\>}H^{1}(M)$ mapping
the sum of a holomorphic and an antiholomorphic form to its cohomology class.
We claim that this map is injective, and, moreover, the form
$\theta:=\alpha^{1,0}-\alpha^{0,1}$ is closed and satisfies
$H^{1}(M)=H^{1,0}(M)\oplus\overline{H^{1,0}(M)}\oplus\langle\theta\rangle=H^{1}(M)$,
that is, $H^{1}(M)$ is generated by cohomology classes of holomorphic and
antiholomorphic forms and $\theta$.

(ii)

Since all holomorphic forms are closed, the antiholomorphic forms are
$\overline{\partial}$-closed and have Dolbeault classes. This gives a natural
embedding $\overline{H^{1,0}(M)}\hookrightarrow H^{0,1}(M)$. We claim that
$H^{0,1}(M)$ is generated by $\overline{H^{1,0}(M)}$ and the Dolbeault class
$A=[\alpha^{0,1}]$, that is, $H^{0,1}(M)=\overline{H^{1,0}(M)}\oplus\langle
A\rangle$.

Proof of (i): Clearly, the natural map
$\partial:\;H^{0,1}(M){\>\longrightarrow\>}H^{1,1}_{BC}(M)$ is injective.
Also, $\overline{H^{1,0}(M)}\subset H^{0,1}(M)$ lies in the kernel of this
map, and its image contains the one-dimensional kernel of the map
$P:\;H^{1,1}_{BC}(M){\>\longrightarrow\>}H^{2}(M)$ (2.1). On the other hand,
$\partial\alpha^{0,1}$ has non-zero degree, because
$\partial\alpha^{0,1}+\overline{\partial\alpha^{0,1}}=\Theta$, hence
$\partial\alpha^{0,1}$ generates $\ker P$.

Proof of (ii): Let $x\in H^{1,0}(M)+\overline{H^{1,0}(M)}$ be an exact 1-form,
$x=df$. Then $dd^{c}f=0$, which is impossible because the map
$f\mapsto\frac{dd^{c}f\wedge\omega}{\omega\wedge\omega}$ is elliptic for any
Hermitian form $\omega$ on $M$. Similarly, for any $x\in H^{1,0}(M)$,
$y\in\overline{H^{1,0}(M)}$, the linear combination $x+y+\theta$ cannot be
exact because in this case the Bott-Chern class of
$\sqrt{-1}\>\Theta=d^{c}\theta=d^{c}(x+y+\theta)$ would vanish, which is
impossible by 2.1 because $\deg\Theta>0$. Finally, $H^{1}(M)$ is generated by
$H^{1,0}(M),\overline{H^{1,0}(M)}$ and $\theta$ because the corresponding
forms generate the Dolbeault cohomology (2.2 (i)), and the Dolbeault
cohomology converges to $H^{1}(M)$ via the Frölicher spectral sequence.

Remark 2.6: As a corollary, we deduce Buchdahl-Lamari theorem (1.1). Indeed,
from 2.2 (ii) it follows that $b_{1}(M)$ is odd for $M$ non-Kähler. This way
one obtains 1.1 from the existence of an exact, positive (1,1)-current on each
complex surface with $b_{1}$ odd (1.1).

###  2.3 Second cohomology of non-Kähler surfaces

Claim 2.7:  Let $M$ be a compact complex surface. Then:

$H^{2,0}(M)=\overline{H^{0,2}(M)}=H^{0}(\Omega^{2}(M)).$

Proof: By Serre’s duality, $H^{0,2}(M)=H^{0}(K_{M})^{*}$, which has the same
dimension as $H^{0}(\Omega^{2}(M))=H^{0}(K_{M})=H^{2,0}(M)$. The natural map
$R:\;H^{2,0}(M){\>\longrightarrow\>}\overline{H^{0,2}(M)}$ is injective,
because its kernel is formed by $\partial$-exact holomorphic forms
$\alpha=\partial\beta$, but for such $\alpha$ one has
$0=\int_{M}d\beta\wedge\overline{\alpha}=\int_{M}\alpha\wedge\overline{\alpha}=\|\alpha\|^{2}$,
which is impossible unless $\alpha=0$. As shown above, these spaces have the
same dimension, hence $R$ is an isomorphism.

Corollary 2.8: ([BHPV, Theorem IV.2.8]) The Hodge-de Rham-Frölicher spectral
sequence of a compact complex surface converges in $E_{1}$.

Proof: We have proven this fact for $H^{1}(M)$ in 2.2. Serre’s and Poincaré
duality give the convergence of this spectral sequence in $H^{3}(M)$. Now it
has to converge in $H^{2}(M)$, because for each instance where $d_{k}\neq 0$,
with $d_{k}:\;E^{p,q}_{k-1}{\>\longrightarrow\>}E^{p-k+1,q+k}_{k-1}$ non-zero,
the group $E^{p,q}_{k-1}$ is replaced by $\ker d_{k}$ and
$E^{p-k+1,q+k}_{k-1}$ by $\operatorname{coker}d_{k}$. Then the total dimension
of the space $\bigoplus_{p+q=d}E^{p,q}_{k-1}$ and
$\bigoplus_{p+q=d}E^{p-k+1,q+k}_{k-1}$ would decrease. However, for $d=1$ and
$d=3$ it is already minimal, hence $d_{k}=0$ for all $k\geqslant 2$.

Remark 2.9: We have shown that $H^{2}(M,{\mathbb{C}})=H^{2,0}(M)\oplus
H^{1,1}(M)\oplus H^{0,2}(M)$ and $H^{2,0}(M)=\overline{H^{0,2}(M)}$, same as
for Kähler surfaces (2.3).

Lemma 2.10:  Let $M$ be a compact complex surface. Then the natural map
$H^{1,1}_{BC}(M){\>\longrightarrow\>}H^{1,1}(M)$ is surjective.

Proof: Let $\alpha\in\Lambda^{1,1}(M)$ be a $\overline{\partial}$-closed form
representing a Dolbeault cohomology class $[\alpha]\in H^{1,1}(M)$. Clearly,
$\alpha$ can be represented by a closed (1,1)-form if and only if the Bott-
Chern class $[\beta]\in H^{2,1}_{BC}(M)$ of $\beta:=\partial\alpha$ vanishes.
Indeed, if $\beta=\partial\overline{\partial}\eta$, then $\alpha-\partial\eta$
is $\overline{\partial}$-closed.

Since the Hodge - de Rham spectral sequence degenerates in $E_{1}$, and
$\beta$ is $\partial$-exact and $\overline{\partial}$-closed, one has
$\beta=\overline{\partial}\gamma$, for some $\gamma\in\Lambda^{2,0}(M)$.
However, the cokernel of the map
$\partial:\;\Lambda^{1,0}(M){\>\longrightarrow\>}\Lambda^{2,0}(M)$ is
generated by the holomorphic 2-forms, and they are all closed, hence
$\overline{\partial}(\Lambda^{2,0}(M))=\overline{\partial}\partial(\Lambda^{1,0}(M))$,
and the Bott-Chern class of $\beta\in\overline{\partial}(\Lambda^{2,0}(M))$
vanishes.

Proposition 2.11:  ([BHPV, Theorem IV.2.14]) Let $M$ be a compact non-Kähler
surface. Then the intersection form on $H^{1,1}(M)$ is negative definite.

Proof: Let $[\alpha]\in H^{1,1}(M)$ be a cohomology class. By 2.3, we can
represent $[\alpha]$ by a closed (1,1)-form $\alpha$. Consider the degree
functional
$\deg_{\omega}:\;H^{1,1}_{BC}(M,{\mathbb{R}}){\>\longrightarrow\>}{\mathbb{R}}$
defined in Subsection 2.1. Since $\deg_{\omega}(\Theta)>0$ for an exact
(1,1)-current $\Theta$, any cohomology class $[\alpha]\in H^{1,1}(M)\subset
H^{2}(M)$ can be represented by a closed (1,1)-form $\alpha$ with
$\deg_{\omega}\alpha=0$. Acting as in the proof of 2.1, we find $f\in
C^{\infty}(M)$ such that $\alpha-dd^{c}f$ is primitive. Replacing $\alpha$ by
$\alpha-dd^{c}f$, we obtain
$\int_{M}\alpha\wedge\alpha=-\|\alpha\|^{2}_{\omega}<0$ as in (2.1).

Remark 2.12: In [Bri2, Theorem 2.37] (see also [BF1, BF2]) it was shown that
the space $H^{1,1}(M)\cap H^{2}(M,{\mathbb{Q}})$ (the rational Neron-Severi
group) of a complex surface with algebraic dimension 0 has negative definite
intersection form.

###  2.4 Vanishing in multiplication of holomorphic 1-forms

2.3 has an interesting corollary.

Proposition 2.13:  Let $M$ be a non-Kähler, compact, complex surface. Then for
any holomorphic 1-forms $\alpha,\beta$, the product $\alpha\wedge\beta$
vanishes, and the product $\alpha\wedge\overline{\beta}$ is exact.

Proof. Step 1: Let $\alpha,\beta$ be holomorphic 1-forms. Then
$\alpha\wedge\beta$ is a holomorphic 2-form, with
$\int_{M}\alpha\wedge\beta\wedge\overline{\alpha}\wedge\overline{\beta}>0$
unless $\alpha\wedge\beta=0$.

Step 2: The intersection form on $H^{1,1}(M)$ is negative definite by 2.3.
Therefore, $\int_{M}\eta\wedge\overline{\eta}=0$ and
$\int_{M}\rho\wedge\overline{\rho}=0$ implies
$\int_{M}\eta\wedge\overline{\rho}=0$ for any $\rho,\eta\in H^{1,1}(M)$. Take
$\eta=\alpha\wedge\overline{\beta}$ and $\rho=\beta\wedge\overline{\alpha}$.
Then clearly
$\int_{M}\eta\wedge\overline{\eta}=\int_{M}\rho\wedge\overline{\rho}=0$, which
implies
$\int_{M}\alpha\wedge\beta\wedge\overline{\alpha}\wedge\overline{\beta}=0$,
hence $\alpha\wedge\beta=0$.

Step 3: To see that $\eta:=\alpha\wedge\overline{\beta}$ is exact we use 2.3
again. Unless $\eta$ is exact, one would have
$\int_{M}\eta\wedge\overline{\eta}<0$, but
$\int_{M}\eta\wedge\overline{\eta}=0$ as we have already shown.

###  2.5 Structure of multiplication in $H^{1}(M)$ for non-Kähler surface
without curves

Further on, we shall need the following lemma.

Lemma 2.14:  Let $M$ be a compact non-Kähler surface, and $\Theta$ an exact
real (1,1)-form. Then there exists a closed form $\theta\in\Lambda^{1}(M)$
such that $d^{c}(\theta)=\Theta$.

Proof: Let $\Theta$ be an exact real (1,1)-form which is non-zero in
$H^{1,1}_{BC}(M)$ and $W$ the space generated by holomorphic and
antiholomorphic forms. Choose a real 1-form $\theta^{c}$ such that
$d\theta^{c}=\Theta$. Since
$\Theta=d((\theta^{c})^{1,0})+d((\theta^{c})^{0,1})$ is real, the form
$\theta:=-I(\theta^{c})$ is closed. By 2.2,
$H^{1}(M)=W\oplus\langle\theta\rangle$.

In this subsection we prove the following structure theorem about
multiplication in $H^{1}(M)$. It is a posteriori true in a general situation,
but we need it only for surfaces without complex curves.

Theorem 2.15:  Let $M$ be a compact non-Kähler surface without curves, and
$\theta\in\Lambda^{1}(M)$ a closed 1-form with integer class such that
$d^{c}(\alpha)$ is non-zero in $H^{1,1}_{BC}(M,{\mathbb{R}})$ (2.5). Denote by
$W\subset\Lambda^{1}(M)$ the subspace generated by holomorphic and
antiholomorphic forms. We identify $W$ with its image in
$H^{1}(M)=W\oplus\langle\theta\rangle$ (2.2). Then

(i)

The multiplication $W\wedge W{\>\longrightarrow\>}H^{2}(M)$ vanishes. The
multiplication $W\wedge\theta{\>\longrightarrow\>}H^{2}(M)$ is injective.

(ii)

Let $\theta^{c}:=I(\theta)$. Then for any non-zero $x\in W$, the form
$x\wedge\theta^{c}$ is closed and represents a non-zero element of $H^{2}(M)$.

(iii)

The Poincaré pairing on the spaces $W\wedge\theta$ and
$W\wedge\theta^{c}\subset H^{2}(M)$ vanishes. The Poincaré pairing between
these two subspaces is non-degenerate.

Proof. Step 1: Multiplication on $W\wedge W{\>\longrightarrow\>}H^{2}(M)$
vanishes by 2.4. Let us prove that the form $x\wedge\theta^{c}$ is closed and
represents a non-zero element of $H^{2}(M)$.

Without restricting the generality, we may assume that $W\neq 0$. We chose a
positive exact current $\Theta:=x\wedge\overline{x}$ for some non-zero 1-form
$x\in W$, and define the closed form $\theta$ such that $d^{c}\theta=\Theta$
as above. Then $d(y\wedge\theta^{c})=y\wedge x\wedge\overline{x}=0$ (2.4), for
any $y\in W$, hence $y\wedge\theta^{c}$ is closed.

Step 2: The injectivity of $W\ext@arrow
0099\arrowfill@\relbar\relbar\longrightarrow{}{x\wedge\theta}H^{2}(M)$ and
$W\ext@arrow
0099\arrowfill@\relbar\relbar\longrightarrow{}{x\wedge\theta^{c}}H^{2}(M)$
would follow if we prove that the Poincaré pairing between these two spaces is
non-degenerate, hence (i) and (ii) follows from (iii).

The Poincaré pairing on $W\wedge\theta$ and $W\wedge\theta^{c}$ vanishes
because $\theta\wedge\theta=\theta^{c}\wedge\theta^{c}=0$. To prove that the
Poincaré pairing between $W\wedge\theta$ and $W\wedge\theta^{c}$ is non-
degenerate, take a holomorphic 1-form $x\in W$. To finish the proof of 2.5
(iii) it would suffice to show that the integral
$\int_{M}\sqrt{-1}\>x\wedge\overline{x}\wedge\theta\wedge\theta^{c}$ is
positive. The form
$\sqrt{-1}\>x\wedge\overline{x}\wedge\theta\wedge\theta^{c}$ is clearly
positive, because it is a product of a $(2,0)$-form and its complex conjugate.
It is non-zero if $x$ is not proportional to the (1,0)-part of $\theta$,
denoted $\theta^{1,0}$.

Step 3: It remains to show that $\theta^{1,0}$ is not proportional to a
holomorphic form; this is where we use that $M$ has no holomorphic curves. In
this case the zeros of any $x\in W$ is a finite set. Suppose that $x\in W$ is
holomorphic and proportional to $\theta^{1,0}$. Then $\theta^{1,0}=\alpha x$,
where $\alpha$ is a smooth function defined outside of the finite set $S$
where $x=0$. Without restricting generality, we may assume that
$d\theta^{1,0}=x\wedge\overline{x}$. Then
$\overline{\partial}\alpha=\overline{x}$, which gives $dd^{c}\alpha=0$ outside
of $S$. Then $\alpha$ is locally the real part of a holomorphic function;
using the Hartogs extension theorem, we obtain that $\alpha$ is smooth and
defined globally on $M$. Then $\alpha=0$ by maximum principle, because
$dd^{c}$ is elliptic.

Corollary 2.16:  Let $M$ be a compact surface without curves, and $W\subset
H^{1}(M,{\mathbb{C}})$ the subspace generated by holomorphic and
antiholomorphic forms. Then $W$ is a rational subspace, that is, there exists
a subspace $W_{\mathbb{Q}}\subset H^{1}(M,{\mathbb{Q}})$ such that
$W=W_{\mathbb{Q}}\otimes_{\mathbb{Q}}{\mathbb{C}}$.

Proof: For $x\in H^{1}(M)$, denote by
$L_{x}:\;H^{1}(M){\>\longrightarrow\>}H^{2}(M)$ the map
$y{\>\longrightarrow\>}x\wedge y$. By 2.5, $\operatorname{rk}L_{\theta}=\dim
W\geqslant 2$ and $\operatorname{rk}L_{x}=1$ for $x\in W$. Therefore $W$ is
the space of all $x\in H^{1}(M)$ such that the $\operatorname{rk}L_{x}=1$.
Since the multiplication in $H^{*}(M)$ is defined over ${\mathbb{Q}}$, the
subspace $W\subset H^{1}(M)$ is rational.

Remark 2.17: 2.5 is true for all surfaces. We leave its proof as an exercise
to the reader.

##  3 Barlet spaces

Barlet spaces are spaces of cycles, that is, closed complex analytic
subvarieties of given dimension in a given complex manifold with
multiplicities (positive numbers) attached to their irreducible components.
They are similar but distinct from the Douady spaces, which are spaces of
closed complex analytic subspaces (possibly with nilpotents in the structure
sheaf).

For more details on Barlet spaces and their properties, please see the book
[BM].

It is important for the sequel that the Barlet space of cycles is complex
analytic, with the topology which is compatible with the (Gromov-)Hausdorff
topology on the set of all closed subvarietiess.

Attached to each Barlet space ${\mathfrak{B}}(M)$ on a manifold $B$ is the
“marked Barlet space” ${\mathfrak{B}}^{m}(M)$, which is the space of pairs
“complex analytic cycle and a point in it”. It is also complex analytic, and
the natural forgetful map $\Psi:\;{\mathfrak{B}}^{m}(M){\>\longrightarrow\>}M$
is also complex analytic. In particular, for any compact irreducible component
$Z$ in the Barlet space, and its marked counterpart $Z^{m}$, the image
$\Psi(Z^{m})$ is a complex analytic subvariety. Geometrically, this means that
for any compact complex-analytic family $Z$ of cycles in $M$, the union of all
these cycles is a complex subvariety in $M$.

Further on, we shall use the following result, which was proven for curves in
[Ve2, Corollary 2.19], and for divisors in [Bar].

Theorem 3.1:  Let $(M,I)$ be a compact complex manifold, and $\omega$ a
Hermitian form. Assume that $dd^{c}(\omega^{k})=0$. Then any connected
component $Z$ of the Barlet space ${\mathfrak{B}}_{k}(M)$ of $k$-cycles in $M$
is compact.

Proof: By Bishop’s theorem ([B]), a Hausdorff limit of a family of compact
complex subvarieties is complex if its Hermitian volume stays bounded. Since
the space of closed subvarieties with Hausdorff topology is compact, this
implies that the set of closed compact $k$-dimensional subvarietes with volume
bounded by a constant $C\in{\mathbb{R}}$ is compact.

Therefore, to prove compactness it would suffice to show that the volume
$\operatorname{Vol}(S)$ is constant as a function of $[S]\in Z$.

Let $X$ denote the marked family associated with $Z$, and
$\pi_{M}:\;X{\>\longrightarrow\>}M$, $\pi_{X}:\;X{\>\longrightarrow\>}Z$ the
forgetful maps. Then the volume function
$\operatorname{Vol}:\;X{\>\longrightarrow\>}{\mathbb{R}}^{>0}$ can be
expressed as $\operatorname{Vol}=(\pi_{X})_{*}\pi_{M}^{*}\omega^{k}$, where
$(\pi_{X})_{*}$ denotes the pushforward of a differential form (generally
speaking, the pushforward not a form, but it is well defined as a current).

Since pullback and pushforward of differential forms commute with $d$,
$d^{c}$, this gives
$dd^{c}\operatorname{Vol}=(\pi_{X})_{*}\pi_{M}^{*}(dd^{c}\omega)$ (see e.g.
[KV, (8.12)], [Ve1, Theorem 2.10] or [I1, Proposition 1.9]). Therefore,
$\operatorname{Vol}$ satisfies $dd^{c}(\operatorname{Vol})=0$ whenever
$dd^{c}(\omega^{k})=0$.

Functions which satisfy $dd^{c}f=0$ are called pluriharmonic. Using local
$dd^{c}$-lemma, it is easy to see that any pluriharmonic function is locally a
sum of a holomorphic and antiholomorphic function.

By Bishop’s theorem, the set $\operatorname{Vol}^{-1}(]-\infty,C])$ is compact
for all $C\in{\mathbb{R}}$, hence $-\operatorname{Vol}$ has a maximum
somewhere in $X$. However, a pluriharmonic function which admits a maximum is
necessarily constant by E. Hopf’s strong maximum principle. Therefore,
$\operatorname{Vol}$ is constant on each connected component of the Barlet
space. Now, each of these components is compact by Bishop.

Applying this result to the space of curves on a complex surface and using the
Gauduchon form $\omega$ (Subsection 2.1), we obtain the following useful
corollary.

Corollary 3.2:  Let $M$ be a compact complex surface, ${\mathfrak{B}}_{1}(M)$
the Barlet space of 1-cycles on $M$ and $Z$ its connected component. Then $Z$
is compact.

##  4 Non-Kähler elliptic surfaces

###  4.1 Elliptic fibrations on non-Kähler surfaces

For a version of the following theorem (with a different proof), see [Bri2,
Theorem 3.17] and [Bri1].

Theorem 4.1:  Let $M$ be a minimal non-Kähler compact complex surface
admitting a continuous family of divisors. Then $M$ admits a holomorphic,
surjective map $\pi:\;M{\>\longrightarrow\>}S$ to a curve, and all fibers of
$\pi$ are elliptic curves, homologous to 0. Moreover, the general fibers of
$\pi$ are isomorphic.

Proof. Step 1: All complex curves $C\subset M$ with the fundamental class
$[C]\neq 0$ have negative self-intersection by 2.3. Therefore, they have
0-dimensional deformation space. Then for any positive-dimensional family of
complex curves in $M$, the curves in this family do not intersect. The general
fibers of this family are homologous curves which do not intersect, hence
their self-intersection is 0\. By 2.3, these curves are homologous to 0.
Denote by ${\mathfrak{B}}_{1}(M)$ the corresponding component of the Barlet
space ${\mathfrak{B}}_{1}(M)$ of 1-dimensional subvarieties on $M$. As shown
in 3, the space ${\mathfrak{B}}_{1}(M)$ is compact. As follows from the
general results on Barlet spaces (see Section 3), the union $Z$ of all curves
in the family parametrized by ${\mathfrak{B}}_{1}(M)$ is complex analytic in
$M$. Since ${\mathfrak{B}}_{1}(M)$ is compact, $Z$ is also compact, hence
$Z=M$. Therefore, this family yields a meromorphic map $\pi$ from $M$ to
${\mathfrak{B}}_{1}(M)$.

If the image of $\pi$ was 2-dimensional, we could choose intersecting curves
in $\tilde{M}$, and they would give a continuous family of intersecting curves
in $M$, which is impossible. Therefore, ${\mathfrak{B}}_{1}(M)$ is a curve.
Taking its closure and possibly blowing up the indeterminacy locus of $\pi$ we
may assume $\pi$ to be homolorphic. Taking the Stein factorization if
necessary and replacing the image of $\pi$ with its normalization, we may
assume that $\pi$ is a holomorphic map, mapping $M$ to a smooth curve $S$ with
connected fibers.

Step 2: Let now $C$ be a general fiber of $\pi$. Using the standard
isomorphism $\pi^{*}(\Omega^{1}S)=N^{*}_{\pi}M$ between the pullback of the
cotangent sheaf and the conormal bundle to the fibers of $\pi$, we obtain an
exact sequence

$0{\>\longrightarrow\>}\pi^{*}(\Omega^{1}S){\>\longrightarrow\>}\Omega^{1}M{\>\longrightarrow\>}\Omega^{1}_{\pi}M{\>\longrightarrow\>}0,$
(4.1)

where $\Omega^{1}_{\pi}M$ is the bundle of holomorphic differentials on fibers
of $\pi$. This gives $K_{M}{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}=K_{C}$,
that is, the canonical bundle to $M$ restricted to $C$ gives the canonical
bundle to $C$. Since $C$ is homologous to 0, $\int_{C}c_{1}(K_{M})=0$, hence
the degree of $K_{C}$ is equal 0. This implies that $C$ is an elliptic curve.

Step 3: Let $\omega_{S}$ be a Kähler form on $S$. Then the (1,1)-form
$\Theta:=\pi^{*}(\omega_{Z})$ is positive, hence its square is non-negative.
By 2.3, the intersection form on $H^{1,1}(M)$ is negative definite, hence
$\Theta$ is exact. Let $\alpha$ be a real 1-form such that
$d(\alpha^{c})=\Theta$, and $\alpha^{1,0}$, $\alpha^{0,1}$ its Hodge
components. Since $\Theta$ is real, the form
$\theta:=-\sqrt{-1}\>\alpha^{1,0}+\sqrt{-1}\>\alpha^{0,1}$ is closed, and
$-I(\theta)=\alpha$, bringing $-d^{c}\theta=\Theta$. For any curve
$C=\pi^{-1}(s)$, one has
$d\theta{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}=d^{c}\theta{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}=0$,
hence the restriction of $\theta^{1,0}$ to $C$ is holomorphic. The form
$\Theta=-d^{c}\theta$ is non-zero in $H^{1,1}_{BC}(M)$ by 2.2, hence $\theta$
is non-exact. Any first cohomology class which vanishes on the fibers of $\pi$
is obtained as a pullback from $S$, which can be seen from the cohomology
exact sequence

$0{\>\longrightarrow\>}H^{1}(S){\>\longrightarrow\>}H^{1}(M){\>\longrightarrow\>}H^{0}(S,{\mathcal{R}}^{1}\pi_{*}({\mathbb{C}}_{M}))$

associated with the Leray spectral sequence,

Since $\theta$ is the only class in $H^{1}(M)$ not lifted from $S$ (2.2), this
implies that the cohomology class of
$\theta{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$ is non-trivial. We obtain
that $\theta^{1,0}{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$ is a non-
trivial holomorphic form.

Now, suppose that $\pi$ has a degenerate fiber $C_{0}$ which does not contain
an elliptic curve. A degeneration of an elliptic curve is either an elliptic
curve with some rational curves attached or a union of rational curves, which
is seen from the general theory of deformations for complex curves ([Ko],
[AL]). Since a rational curve does not have non-zero holomorphic
differentials, one has
$\theta{\left|{}_{{\phantom{|}\\!\\!}_{C_{0}}}\right.}=0$.

The fibers $C_{0}$ of $\pi$ are deformation retracts of general fibers (see
[Mor], [Pe], [C]). Therefore, any closed form on $M$ which vanishes on
$\pi_{1}(C_{0})$ represents 0 in $H^{1}(C)$ for a general fiber $C$. Applying
this to $\theta$, we arrive at contradiction, because the cohomology class of
$\theta{\left|{}_{{\phantom{|}\\!\\!}_{C}}\right.}$ is non-trivial.

Step 4: We are going to prove that all fibers of $\pi$ are elliptic curves,
while a priori a singular fiber might be a union of an elliptic curve and a
collection of rational curves.

Consider an irreducible curve $D\subset M$. Then $D$ lies in a fiber of $\pi$.
Either $D$ has self-intersection 0 (and then it is an elliptic curve as the
above argument shows), or $D$ has negative self-intersection. The singular
fiber of $\pi:\;M_{1}{\>\longrightarrow\>}M$, being a deformation of an
elliptic curve, is either a union of rational curves or a union of a rational
curve and a torus, with the graph of incidence of irreducible components being
a tree (see [AL, Theorem VII.2.2.1]). The first case is easily excluded by the
reasoning in the previous step. Blowing up all points where the components
have multiple intersections, we may assume that they have simple
intersections. For each $D$, denote by $D_{1}$ the union of the rest of the
irreducible components of the corresponding fiber $\pi^{-1}(S)$. Then the
self-intersection of $D+D_{1}$ is zero, $(D+D_{1})^{2}=0$, and $D_{1}=-D$,
giving $D^{2}=2(D,D_{1})-D^{2}$, hence $D^{2}=-(D,D_{1})$. All components of
$\pi^{-1}(S)$ are elliptic or rational curves, and its incidence graph is a
tree, hence the outermost rational curve $D$ has simple intersection with its
complement $D_{1}$, and we have $D^{2}=-1$. Then we can blow down this curve
and the rest of rational curve components one by one, arriving at a smooth
manifold obtained from $M$ by blowing down all rational curves in the special
fibers of $\pi$. Since $M$ is minimal, this is impossible, unless all fibers
of $\pi$ are elliptic.

Step 5: We have shown that all fibers of $\pi$ are elliptic curves, with
singularities in multiple fibers. Now we prove that the general fibers are
isomorphic. Let $s\in S$ be a point corresponding to a multiple fiber, and $U$
its small neighbourhood. Then $\pi$ becomes a smooth fibration after passing
to a finite ramified cover $U^{\prime}{\>\longrightarrow\>}U$, with
$U=U^{\prime}/G$, where $G$ is a finite group (4.2). The correspondng map
$U^{\prime}{\>\longrightarrow\>}A_{1}$ to the coarse moduli space $A_{1}$ of
elliptic curves is $G$-invariant, hence it is well defined on $U$. This allows
to extend the map to the coarse moduli to the whole $S$, obtaining a
holomorphic map $\tau:\;S{\>\longrightarrow\>}A_{1}$. Since $S$ is compact,
and $A_{1}$ is a non-compact curve, $\tau$ is constant. Therefore, the general
fibers of $\pi$ are all isomorphic.

Corollary 4.2:  Let $M$ be a non-Kähler, compact, complex surface with
$b_{1}(M)>3$. Then $M$ admits an elliptic fibration.

Proof: By 2.2, $\dim H^{1,0}(M)=\frac{b_{1}(M)-1}{2}$, hence $\dim
H^{0}(\Omega^{1}(M))\geqslant 2$. By 2.4, all globally defined holomorphic
1-forms on $M$ are pointwise proportional. Consider the rank 1 sheaf
$L\subset\Omega^{1}(M)$ generated by holomorphic 1-forms. If $\dim
H^{1,0}(M)=\dim H^{0}(L)>1$, the zero divisors of the sections of this sheaf
form a continuous family of curves on $M$, and 4.1 can be applied.

Remark 4.3: The result we have proven does not imply that the map $\pi$ is a
submersion. Indeed, $\pi$ may have multiple fibers. For example, take the
product $M={\mathbb{C}}P^{1}\times T^{2}$, where $T^{2}$ is an elliptic curve.
Consider an order 2 automorphism
$\sigma:\;{\mathbb{C}}P^{1}{\>\longrightarrow\>}{\mathbb{C}}P^{1}$, mapping
$[x:y]$ to $[x:-y]$. Let $v\in T^{2}$ be an element of order 2, and let $\tau$
act on $M$ by mapping $(x,t)$ to $(\sigma(x),t+v)$. Clearly, $\tau$ acts on
$M$ without fixed points, and the quotient $M/\tau$ is projected to
${\mathbb{C}}P^{1}/\sigma$ with multiple fibers of order 2 in points $[1:0]$
and $[0:1]$. This construction gives a Kähler elliptic surface, but it is
possible to perform a similar construction for an elliptic surface which is
not Kähler (for example, a quotient of a primary Kodaira surface gives a
secondary Kodaira surface, fibered over ${\mathbb{C}}P^{1}$ with multiple
fibers).

###  4.2 Isotrivial elliptic fibrations

Proposition 4.4:  Let $M$ be a be a non-Kähler compact complex surface,
admitting an elliptic fibration $\pi:\;M{\>\longrightarrow\>}S$. Denote by $C$
a general fiber of $M$. Then there is a natural holomorphic action of $C$ on
$M$, transitive on fibers and free on non-multiple fibers.

Proof: Two pairs of points $(a,b)$ and $(a_{1},b_{1})$ on a smooth genus 1
curve $T$ are called rationally equivalent if there exists an translation
automorphism of $T$ mapping $(a,b)$ to $(a_{1},b_{1})$. Clearly, the set
$\operatorname{Jac}(T)$ of such pairs up to rational equivalence is a curve
isomorphic to $T$, with a fixed point $(x,x)$, which we can consider as the
group unit. We can consider $T$ as a torsor over the group
$\operatorname{Jac}(T)$. 111A torsor over a group $G$ is a set with a free and
transitive $G$-action.

Let $\operatorname{Jac}(M)$ be the fiberwise Jacobian of $M$, that is, the set
of pair of points $x,y\in C_{s}=\pi^{-1}(s)$ up to rational equivalence. The
map $\pi:\;\operatorname{Jac}(M){\>\longrightarrow\>}S$ is a locally trivial
fibration with a globally defined section, hence it is trivial. The natural
fiberwise action of $\operatorname{Jac}(M)=C\times S$ on $M$ defines the
action on $M$, transitive on fibers and free on general fibers.

Definition 4.5: Let $M$ be a complex manifold equipped with an action of a
compact complex torus $T$, with the fibers of the same dimension, and
$\pi:\;M{\>\longrightarrow\>}S$ the quotient map. Then
$\pi:\;M{\>\longrightarrow\>}S$ is called an isotrivial toric fibration, and
an isotrivial elliptic fibration if $\dim_{\mathbb{C}}T=1$.

Remark 4.6: In some literature, “isotrivial elliptic fibration” is a fibration
with all smooth fibers isomorphic to the same elliptic curve and the only
singularities in multiple fibers. In [Bri2] and [Bri1], the same notion is
called “a quasi-bundle”.

Remark 4.7:  Let $\pi:\;M{\>\longrightarrow\>}S$ be an isotrivial elliptic
fibration over a curve $S$, with fiber $C$. We associate wih $\pi$ the
following orbifold structure on $S$. For any multiple fiber $R=\pi^{-1}(s)$ of
$\pi$, consider the group $\Gamma_{R}\subset C$, obtained as the kernel of the
natural action of $C$ on $R$. Taking the quotient $M/\Gamma_{R}$, we obtain
another fibration, which is locally trivial in a neighbourhood of $s$. A
smooth section of this fibration in a neighbourhood of $s$ gives a
$\Gamma_{R}$-invariant multisection $\tilde{U}{\>\longrightarrow\>}U$ of
$\pi:\;\pi^{-1}(U){\>\longrightarrow\>}U$. Then $U$ is obtained as a finite
quotient of $\tilde{U}$, and the lifting of $\pi$ to $\tilde{U}$ is locally
trivial.

Remark 4.8: We obtained that any isotrivial elliptic bundle
$\pi:\;M{\>\longrightarrow\>}S$ over a curve defines an orbifold structure on
this curve, and $\pi$ is locally trivial in this orbifold structure. The same
argument would work for any base $S$ regardless of dimension.

Remark 4.9: To simplify the notation and the arguments, we shall deal with the
smooth orbifold fibrations in the same way as for the smooth ones, and use the
standard terminology instead of adding “orbifold smooth” everywhere. Many
(most) of the standard results and constructions in smooth category are
extended to the orbifold category in the usual way.

The topology and geometry of isotrivial toric fibrations and principal toric
fibrations was explored in some depth in [Hö]. For our present purposes we
shall need the following theorem. Notice that the isotrivial toric fibration
can be considered as a locally trivial principal toric fibration if we work in
the orbifold category. Following [Hö], we define the Chern classes associated
with isotrivial toric fibrations as follows.

Theorem 4.10:  Let $T(S)$ be the sheaf of $T$-valued smooth functions on $S$.
Let $\tilde{T}={\mathbb{R}}^{n}$ the universal cover of $T$, and denote by
$\tilde{T}(S)$ the sheaf of smooth $\tilde{T}$-valued functions on $S$. Denote
by ${\mathbb{Z}}^{n}(S)$ the constant sheaf with fiber ${\mathbb{Z}}^{n}$.
Then the following exact sequence of sheaves

$0{\>\longrightarrow\>}{\mathbb{Z}}^{n}(S){\>\longrightarrow\>}\tilde{T}(S){\>\longrightarrow\>}T(S){\>\longrightarrow\>}0$

gives an exact sequence in cohomology

$0=H^{1}(\tilde{T}(S)){\>\longrightarrow\>}H^{1}(TS){\>\longrightarrow\>}H^{2}({\mathbb{Z}}^{n}(S)){\>\longrightarrow\>}H^{1}(\tilde{T}(S))=0$

and a bijective correspondence between the set $H^{1}(T(S))$ of principal
toric fibrations and the cohomology group
$H^{2}({\mathbb{Z}}^{n}(S))=H^{2}(M,{\mathbb{Z}})^{n}$.

Proof: Clear (see [Hö] for details).

Definition 4.11: Let $\pi:\;M{\>\longrightarrow\>}S$ be an isotrivial toric
fibration, with $T={\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$ its fiber. The $n$
cohomology classes associated to $\pi$ as in 4.2 are called the Chern classes
of the isotrivial toric fibration.

Remark 4.12:  In the case when $\pi:\;M{\>\longrightarrow\>}S$ is an
isotrivial elliptic fibration over a compact curve, one has
$H^{2}(S,{\mathbb{Z}})={\mathbb{Z}}$. Therefore a toric fibration is uniquely
determined by the vector
$c_{1}(\pi)\in{\mathbb{Z}}^{2}=H^{2}(S,{\mathbb{Z}})^{2}$. Recall that a
vector $v$ in a lattice ${\mathbb{Z}}^{n}$ is called primitive if $v$ is not
divisible by an integer $n>1$. For any primitive vector $v$ in a lattice
$\Lambda={\mathbb{Z}}^{2}$, there exists $w\in\Lambda$ such that
$\Lambda=\langle w,v\rangle$. Therefore, the Chern classes of an isotrivial
elliptic fibration over a curve are determined (up to an automorphism of
$\Lambda$) by the biggest $n\in{\mathbb{Z}}$ such that $c_{1}(\pi)$ is
divisible by $n$. Using this observation, we shall understand $c_{1}(\pi)$ as
a non-negative integer; it is equal to zero when $v=0$ and equal to the
largest integer divisor of $v$ when $v\neq 0$.

Now we shall prove that an isotrivial elliptic fibration
$\pi:\;M{\>\longrightarrow\>}S$ over a curve with $c_{1}(\pi)\neq 0$ is always
obtained as a quotient of the total space of an ample
${\mathbb{C}}^{*}$-bundle. We need the following preliminary topological
statement.

Claim 4.13:  Let $\pi:\;M{\>\longrightarrow\>}S$ be an isotrivial elliptic
fibration over a compact curve $S$, and $C$ its general fiber. Assume that
$c_{1}(\pi)\neq 0$. Then the rank of the natural map
$H_{1}(C){\>\longrightarrow\>}H_{1}(M)$ is 1. Moreover, there exists a normal
subgroup $G\subset\pi_{1}(M)$ with $\pi_{1}(M)/G={\mathbb{Z}}$ such that the
corresponding ${\mathbb{Z}}$-cover is infinite on $C$.

Proof: Consider the exact sequence

$H_{2}(S)\stackrel{{\scriptstyle\delta}}{{{\>\longrightarrow\>}}}H_{1}(C)\stackrel{{\scriptstyle\psi}}{{{\>\longrightarrow\>}}}H_{1}(M){\>\longrightarrow\>}H_{1}(S){\>\longrightarrow\>}0$
(4.2)

obtained from the Leray spectral sequence of the fibration. It is easy to see
that $\delta$ is dual to the Chern class of $\pi$. Therefore, the map
$\psi:\;H_{1}(C){\>\longrightarrow\>}H_{1}(M)$ has rank 1 when $c_{1}(\pi)\neq
0$ and 2 when $c_{1}(\pi)=0$. Therefore, there exists an element
$v\in\pi_{1}(C)$ such that its image in $\pi_{1}(M)$ has infinite order. Take
a homomorphism $H_{1}(M){\>\longrightarrow\>}{\mathbb{Z}}$ which is non-zero
on $\psi(H_{1}(C))$. Since
$H_{1}(M)=\frac{\pi_{1}(M)}{[\pi_{1}(M),\pi_{1}(M)]}$, the map $v$ defines a
group homomorphism $\pi_{1}(M){\>\longrightarrow\>}{\mathbb{Z}}$ which is non-
trivial on $\pi_{1}(C)=H_{1}(C)$.

Proposition 4.14:  Let $\pi:\;M{\>\longrightarrow\>}S$ be an isotrivial
elliptic fibration over a compact curve $S$, and $C$ its general fiber.
Consider a ${\mathbb{Z}}$-cover $\varphi:\;\tilde{M}{\>\longrightarrow\>}M$
such that $\varphi^{-1}(C){\>\longrightarrow\>}C$ is an infinte cover. Then
$\tilde{M}$ is the total space of a principal $C^{*}$-bundle associated with a
line bundle $L$ on $M$, and $M$ is obtained as the quotient of
$\operatorname{Tot}(L^{\circ})$ of the total space of non-zero vectors in $L$
by a holomorphic automorphism acting acting on the base $S$ as
$\varphi:\;S{\>\longrightarrow\>}S$, with $\psi$ of finite order
$\varphi^{\circ}(L)\simeq L$, and on the fibers as an isomorphism
$\lambda:\;L{\>\longrightarrow\>}\varphi^{*}L$. Finally, $|c_{1}(L)|$ is equal
to $c_{1}(\pi)$.

Proof: Since $\varphi^{-1}(C){\>\longrightarrow\>}C$ is an infinite cover, the
space $\tilde{M}$ is a ${\mathbb{C}}^{*}$-bundle over $S$. Denote the
associated vector bundle by $L$. Then $\tilde{M}=\operatorname{Tot}(L^{*})$,
and $M$ is obtained from $\operatorname{Tot}(L^{\circ})$ as a quotient by an
automorphism $\gamma$ commuting with the ${\mathbb{C}}^{*}$-action. However,
all automorphisms of ${\mathbb{C}}^{*}$ commuting with the
${\mathbb{C}}^{*}$-action are multiplication by a number. Therefore, $\gamma$
gives a holomorphic section $\lambda\in\operatorname{Hom}(L,\varphi^{*}(L))$,
where $\varphi$ denotes the action of $\gamma$ on $C$. The fibers of $\pi$ are
compact, hence $\varphi$ has to be of finite order.

The Chern class of $L$ is obtained from the $U(1)$-bundle associated with $L$,
and this bundle is homotopy equivalent to $\tilde{M}$, hence $c_{1}(L)$ is
equal to $c_{1}$ of the circle bundle associated with the
${\mathbb{C}}^{*}$-bundle $\tilde{M}{\>\longrightarrow\>}S$. However, $c_{1}$
of $\tilde{M}{\>\longrightarrow\>}S$ is by construction equal with $\pm
c_{1}(\pi)$ (the $\pm$ sign is due to the ambiquity of the definition of
$c_{1}$ of the toric fibration, see 4.2).

###  4.3 Blanchard theorem and LCK structure on non-Kähler elliptic surfaces

The main result of the present section is the following theorem, proven at the
end of this subsection.

Theorem 4.15:  ([Bel, Theorem 1]) Let $M$ be a compact, non-Kähler, minimal
surface admitting an elliptic fibration. Then $M$ is Vaisman.

We use Blanchard theorem (see [Bl] or [R]), which is applied to the present
case as follows.

Theorem 4.16:  Let $\pi:\;M{\>\longrightarrow\>}S$ be an isotrivial elliptic
fibration. Then $M$ is Kähler if and only if $c_{1}(\pi)=0$.

Proof: Suppose that $c_{1}(\pi)=0$, and consider the exact sequence

$0{\>\longrightarrow\>}H^{1}(S){\>\longrightarrow\>}H^{1}(M){\>\longrightarrow\>}H^{1}(C)\stackrel{{\scriptstyle\delta^{*}}}{{{\>\longrightarrow\>}}}H^{2}(S)$

dual to (4.2). As shown in the proof of 4.2, one has $\delta^{*}=0$ if and
only if $c_{1}(\pi)=0$. However, if $\delta^{*}\neq 0$, the pullback
$\pi^{*}(\omega_{S})$ of the volume form is exact, which is impossible when
$M$ is Kähler by 1.1.

Assume now that $c_{1}(\pi)=0$, equivalently, that $\delta^{*}=0$. Using 4.2,
we obtain $M$ as a quotient of a total bundle $\operatorname{Tot}(L^{\circ})$
by an automorphism acting on the base $S$ as
$\varphi:\;S{\>\longrightarrow\>}S$ and on the fibers as a constant
endomorphism $\lambda:\;L{\>\longrightarrow\>}\varphi^{*}L$, $|\lambda|\neq
0$. Since $c_{1}(L)=\pm c_{1}(\pi)=0$, this bundle admits a flat Hermitian
connection. Denote by $\rho$ its monodromy action. Since $\rho$ acts on the
fibers of $L$ isometrically, we can chose a $\rho$-invariant Kähler metric
$\omega_{F}$ on the fibers of $L^{\circ}/\lambda$, which are elliptic curves.
Extending $\omega_{F}$ to $M$ using the flat Ehresmann connection on
$\pi:\;M{\>\longrightarrow\>}S$, we obtain a closed, positive (1,1)-form
$\hat{\omega}_{F}$ on $M$ which is strictly positive on the fibers of $\pi$;
the sum $\omega:=\hat{\omega}_{F}+\pi^{*}(\omega_{S})$ is a positive, closed,
strictly positive almost everywhere (1,1)-form. Then $\omega^{2}>0$, in
contradiction with the negativity of the intersection form on $H^{1,1}(M)$
(2.3).

Let us prove 4.3 (see also [Va2, Theorem 3.5 and the subsequent Remark] for
the regular case, and also [Vu2]). If $M$ is non-Kähler, it is obtained as a
quotient of $\operatorname{Tot}(L^{\circ})$ by
$\lambda:\;L{\>\longrightarrow\>}\varphi^{*}L$, where $c_{1}(L)\neq 0$ by 4.3.
Replacing $L$ by its dual bundle if necessary, we may assume that $L$ is
ample. Choose a Hermitian structure on $L$ such that its curvature is a
positive (1,1)-form on $S$, and let $\psi\in
C^{\infty}(\operatorname{Tot}(L^{\circ}))$ be the function $\psi(v)=|v|^{2}$.

We want the isomorphism $\lambda:\;L{\>\longrightarrow\>}\varphi^{*}L$ to have
constant length. Since $\varphi$ is of finite order, we can always represent
the curvature $\Theta$ of $L$ by a $\varphi$-invariant form. Using the
$dd^{c}$-lemma as usual, we choose a metric on $L$ with curvature $\Theta$.
Then the curvature of $\operatorname{Hom}(L,\varphi^{*}L)$ is
$\varphi^{*}(\Theta)-\Theta=0$. Therefore,
$\operatorname{Hom}(L,\varphi^{*}L)$ is a flat unitary bundle; any section
$\lambda\in\operatorname{Hom}(L,\varphi^{*}L)$ gives a holomorphic section of
a flat unitary bundle, hence it is of constant length.

Then $dd^{c}(\psi)$ is a Kähler form on $\operatorname{Tot}(L^{\circ})$ ([Bes,
(15.19)]). Then $\lambda$ acts on $\operatorname{Tot}(L^{\circ})$ by
holomorphic homotheties, hence the quotient
$M=\operatorname{Tot}(L^{\circ})/\langle\lambda\rangle$ is LCK. Also, this
quotient is Vaisman, because the standard action of ${\mathbb{C}}^{*}$ on
$\operatorname{Tot}(L^{*})=\tilde{M}$ is by holomorphic homotheties, [KO].

##  5 Class VII surfaces

In this section we prove the following theorem.

Theorem 5.1:  Let $M$ be a non-Kähler compact minimal complex surface. Then
$M$ is isotrivial elliptic or belongs to class VII (or both).

Proof. Step 1: As $b_{1}(M)>0$, and
$H_{1}(M,{\mathbb{Z}})=\pi_{1}(M)/[\pi_{1}(M),\pi_{1}(M)]$, there exists a
subgroup of any given finite index $r$ in $M$. Denote the corresponding cover
by $\sigma:\;M_{1}{\>\longrightarrow\>}M$. Then $M_{1}$ is a compact manifold.
Since $\chi({\cal O}_{M})=\frac{c_{1}^{2}+c_{2}}{12}$ is expressed through the
curvature, one has $\chi({\cal O}_{M_{1}})=d\chi({\cal O}_{M})\geqslant d$ for
any $d$-sheeted cover $\sigma:\;M_{1}{\>\longrightarrow\>}M$. Unless
$\chi(M)=0$, we can find a cover $M_{1}$ with $\chi(M_{1})<-3$ or
$\chi(M_{1})>3$. In the first case, $M_{1}$ is elliptic by 4.1. In the second
case, the canonical bundle $K_{M_{1}}$ satisfies $\dim
H^{0}(K_{M_{1}})\geqslant 2$, hence it has sections which give a continuous
family of divisors. Therefore, for $\chi({\cal O}_{M})\neq 0$, the finite
cover $M_{1}$ has a continuous family of divisors. Then $M$ also has such a
continuous family, and it is elliptic by 4.1.

Step 2: Assume that $b_{1}(M)=3$, but $M$ is not elliptic. Then $\chi(M)=0$,
hence $h^{0,1}=2$ and $h^{0,2}=1$, and the same is true for all non-ramified
covers of $M$. We are going to prove that $M$ contains no curves.

Let $\nu$ be a non-zero section of $K_{M}$ and $C$ the zero divisor of $\nu$.
If $C$ has a component $D$ with negative self-intersection, the adjunction
formula $ND=KM{\left|{}_{{\phantom{|}\\!\\!}_{D}}\right.}=KD\otimes N^{*}D$
implies that $KD=ND\otimes ND$, hence $D$ is a rational curve with self-
intersection -1, which is impossible because $M$ is minimal. This implies that
the self-intersection of $C$ is non-negative; by 2.3, $C$ is homologous to 0,
and $c_{1}(M)^{2}=0$. This gives $0=\chi({\cal
O}_{M})=\frac{c_{1}^{2}+c_{2}}{12}=\frac{c_{2}}{12}$, hence $b_{2}(M)=6-2=4$.

Now we can show that $M$ has no curves with negative self-intersection;
indeed, if $M$ has $r$ such curves, a non-ramified $d$-cover $M_{1}$ would
have $dr$ curves, which would imply $b_{2}(M_{1})\geqslant dr$, contradicting
$b_{2}(M)=b_{2}(M_{1})=4$.

Consider curves $C_{1},...,C_{n}\subset M$ of self-intersection 0 (and hence
homologous to 0), and let $D$ be their union. Using the exact sequence

$0{\>\longrightarrow\>}{\cal O}(-D){\>\longrightarrow\>}{\cal
O}_{M}{\>\longrightarrow\>}\bigoplus_{i}{\cal
O}_{C_{i}}{\>\longrightarrow\>}0$

and the corresponding long exact sequence

${\mathbb{C}}=H^{0}({\cal
O}_{M}){\>\longrightarrow\>}{\mathbb{C}}^{n}=\bigoplus_{i}H^{0}({\cal
O}_{C_{i}}){\>\longrightarrow\>}H^{1}({\cal O}(-D)),$

we obtain that $\dim H^{1}(L)\geqslant n-1$, where $L:={\cal O}(-D)$. However,
$\chi(L)=\chi({\cal O}_{M})$ because $c_{1}(L)=[C]=0$, and $\dim H^{2}(L)=\dim
H^{0}(K_{M}\otimes L^{*})\leqslant 1$, hence $\dim H^{1}({\cal
O}(-D))\leqslant 2$. This implies that $n\leqslant 3$. Since
$b_{1}(M)>b_{1}(C)$, there exists an order $d$ covering $M_{1}$ of $M$ which
is trivial on $C$, obtained by the same argument as used in Step 1, hence any
curve with self-intersection 0 gives a rise to $d$ such curves on $M_{1}$.
This is impossible, because, as we just showed, $M_{1}$ cannot have more than
3 non-intersecting curves.

Step 3: We can prove now that all surfaces with $b_{1}(M)=3$ and $\chi({\cal
O}_{M})=0$ are elliptic. Arguing by absurd, we may assume $M$ is non-elliptic,
but in this case $M$ has no complex curves (Step 2). Consider the space
$W\subset H^{1}(M)$ generated by holomorphic and antiholomorphic forms. Let
$\alpha\in\Lambda^{1,0}(M)$ be a holomorphic 1-form generating the space
$H^{1,0}(M)$. Fix $x\in M$ and define the Albanese map by taking $y\in M$ to
$\int_{\gamma}\alpha$, where $\gamma$ is a path connecting $x$ to $y$. This
map depends on the choice of the path $\gamma$, hence it gives a map
$\operatorname{Alb}:\;M{\>\longrightarrow\>}{\mathbb{C}}/\Lambda$, where
$\Lambda\subset{\mathbb{C}}$ is the period lattice of $\alpha$, that is, the
set $\\{\int_{v}\alpha\ \ |\ \ v\in H_{1}(M,{\mathbb{Z}})\\}$. By 2.5, $W$ is
rational. Therefore, the integral $\int_{v}\alpha$ vanishes on one of the
generators in $H_{1}(M,{\mathbb{Z}})$, and $\Lambda$ is a rank 2 lattice in
${\mathbb{C}}$. It is easy to see that it is discrete. Then the Albanese map
gives a fibration $M{\>\longrightarrow\>}{\mathbb{C}}/\Lambda$, and $M$
contains a continuous family of curves, hence it is elliptic.

Step 4: It remains to show that $M$ is of class VII or elliptic if $\chi({\cal
O}_{M_{1}})=0$. When $b_{1}(M)=3$, the surface $M$ is elliptic (Step 3).
Therefore, we may assume that $b_{1}(M)=h^{0,1}(M)=1$ and the same is true for
all covers $M_{1}$ of $M$, ramified or unramified. Unless $M$ is elliptic, we
have $\chi(M)=0$, hence $h^{0,1}(M)=0$. To prove that $M$ is class VII one has
to check that $H^{0}(K_{M}^{n})=0$ for any $n>0$.

If $K^{n}_{M}$ admits a non-zero section, then $K_{M_{1}}$ admits a non-zero
section for some finite ramified cover $\sigma:\;M_{1}{\>\longrightarrow\>}M$.
Indeed, given a section $\alpha$ of $K^{n}_{M}$, with zero divisor $C$,
locally around a point $x\in M\backslash C$, one can take its $n$-th degree
root $\sqrt[n]{\alpha}$ and obtain a section of $K_{M}$. The sheaf of such
sections is finite and locally constant on $M\backslash C$, hence it becomes
trivial after passing to a finite cover $\widetilde{M\backslash
C}{\>\longrightarrow\>}M\backslash C$. In a neighbourhood $U$ of $x\in C$, we
interpret $\sqrt[n]{\alpha}$ as a multivalued function with values in $K_{M}$,
which can be trivialized in $U$. The graph of $\sqrt[n]{\alpha}\subset U\times
C$ can be glued to $\widetilde{M\backslash C}$, giving a ramified cover
$\sigma:\;M_{1}{\>\longrightarrow\>}M$ and a section of $K_{M_{1}}$. Then
$h^{0,2}(M_{1})\neq 0$, hence $M_{1}$ is elliptic (Step 3). Therefore,
$M=\sigma(M_{1})$ contains a continuous family of holomorphic curves; it is
elliptic by 4.1. Isotriviality of the elliptic fibration follows from 4.2.

##  6 Brunella’s theorem: all Kato surfaces are LCK

The Kato surfaces are also called “Global spherical shell surfaces”. By
definition a Kato surface is a surface $M$ which possesses a global spherical
shell, that is, an open subset $U\subset M$ which is biholomorphic to a
neighbourhood of $S^{3}$ in ${\mathbb{C}}^{2}$ and such that $M\backslash U$
is connected. In this section we prove Brunella’s theorem, showing that all
Kato surfaces are locally conformally Kähler.

For the original definition and early works on Kato surfaces see [Ka1, Ka2,
Ka3, Dl], and for Brunella’s original proof, [Bru1] and [Bru2].

It is not hard to see that all Kato surfaces are deformations of a blow-up of
a Hopf surface. To see this, we consider the following explicit construction
of Kato surfaces.

Let $M$ be a Kato surface, and $S\subset U\subset M$ the corresponding
3-sphere. Consider the map $\chi:\;\pi(M){\>\longrightarrow\>}{\mathbb{Z}}$
mapping a path $\gamma$ to the intersection index $\gamma\cap S$. Clearly,
$\chi$ is a group homomorphism. Denote the corresponding ${\mathbb{Z}}$-cover
by $\tilde{M}$, and the preimages of $S$ in $\tilde{M}$ by $S_{i}$,
$i\in{\mathbb{Z}}$ (these preimages can be enumerated, because the deck
transform group ${\mathbb{Z}}$ acts on the set of preimages of $S$ freely and
transitively). Denote by $M_{i}$ the subsets of $\tilde{M}$ situated between
$S_{i}$ and $S_{i-1}$. Clearly, each $M_{i}$ is a fundamental domain of the
deck transform action.

Action of ${\mathbb{Z}}$ on $\tilde{M}$, spherical shells and fundamental
domains

Each $M_{i}$ has two boundary components, with $S_{i}$ pseudo-convex and
$S_{i-1}$ pseudo-concave. Gluing a ball $B$ to $S_{i-1}$, we obtain a manifold
$\hat{M}_{i}=B\coprod_{S_{i-1}}M_{i}$ which is compact and has strictly
pseudoconvex boundary. By the solution of Levi problem (Oka, Hörmander),
$\hat{M}_{i}$ is holomorphically convex. Using Remmert reduction theorem [Re],
we obtain a proper map with connected fibers
$p:\;\hat{M}_{i}{\>\longrightarrow\>}X$, where $X$ is Stein. Since the
neighbourhood of the boundary of $X$ is biholomorphic to a neighbourhood of
$S^{3}\subset B$, Hartogs theorem implies that $X$ is in fact biholomorphic to
$B$. Then $p:\;\hat{M}_{i}{\>\longrightarrow\>}X$ is a bimeromorphic,
holomorphic map to a ball, and is obtained by a sequence of a blow-ups.

Following Dloussky [Dl], we define the Kato data on a closed ball
$B\subset{\mathbb{C}}^{2}$ as a bimeromorphic, holomorphic map
$\hat{B}{\>\longrightarrow\>}B$, together with an open subset
$B_{0}\subset\hat{B}$ and a biholomorphism from $B_{0}$ to an open ball. Then
the complement $\hat{B}\backslash B_{0}$ has two smooth boundary components,
which are both isomorphic to $S^{3}\subset B$ and can be glued together to
obtain a compact complex surface.

We have just proved the following theorem, originally due to Ma. Kato.
([Ka1]).

Theorem 6.1: Let $M$ be a Kato surface, and $S\subset M$ its global spherical
shell. Then $M$ can be obtained from the Kato data by gluing two boundary
components of $\hat{B}\backslash B_{0}$ as above.

Remark 6.2:  Using the same arguments, it is possible to show that the Kato
surface $M$ is not minimal unless $\hat{B}$ is obtained by blowing up a point
$x_{0}\in B$, then blowing up points which lie on exceptional divisors. Since
a blow-up of an LCK manifold is LCK, we can always assume that $M$ is minimal
and $\hat{B}$ is obtained from $B$ by successive blow-ups in $0\in B$.

Remark 6.3:  Suppose that the Kato data satisfy the assumptions in 6. Then in
the construction of the Kato surface $M$ from the Kato data, we can always
replace the open ball $B$ of radius 1 by a ball $B(r)\subset B$ of radius $r$
and the ball $B_{0}$ by its image $\Psi(B(r))$. Clearly, the resulting Kato
surface is biholomorphic to $M$. Similarly, the ball $B$ can be replaced by
any holomorphically convex domain $U\subset{\mathbb{C}}^{2}$ with smooth
boundary containing the origin $0$.

Definition 6.4: Choose $\varepsilon>0$, and let
$\max_{\varepsilon}:\;{\mathbb{R}}^{2}{\>\longrightarrow\>}{\mathbb{R}}$ be a
smooth, convex function, monotonous in both variables, which satisfies
$\max_{\varepsilon}(x,y)=\max(x,y)$ whenever $|x-y|>\varepsilon$. Then
$\max_{\varepsilon}$ is called a regularized maximum ([D1]). It is easy to see
that the regularized maximum of two plurisubharmonic functions is again
plurisubharmonic. This construction allows one to “glue” plurisubharmonic
functions and the corresponding Kähler metrics.

Now we can prove Brunella’s theorem.

Theorem 6.5:  Let $M$ be a Kato surface. Then $M$ is locally conformally
Kähler.

Proof. Step 1: Let $\pi:\;\hat{B}{\>\longrightarrow\>}B$ and $B_{0}\subset B$
be the Kato data. Denote by $\Psi:\;B{\>\longrightarrow\>}B_{0}$ the
corresponding biholomorphic equivalence. Choose a Kähler metric $\hat{\omega}$
on $\hat{B}$.

Brunella’s theorem is proved by finding a metric $\hat{\omega}$ on $\hat{B}$
with the following automorphic condition. Consider the space $\tilde{M}$
obtained by gluing ${\mathbb{Z}}$ copies of $\hat{B}\backslash B_{0}=M_{i}$ as
above. A Kähler metric $\tilde{\omega}$ on $\tilde{M}$ is called
${\mathbb{Z}}$-automorphic if the deck transform group mapping $M_{i}$ to
$M_{j}$ acts on $(\tilde{M},\tilde{\omega})$ by homotheties. To obtain such a
form we need to find a Kähler form $\hat{\omega}$ on $\hat{B}$ such that
$\hat{\omega}{\left|{}_{{\phantom{|}\\!\\!}_{B_{0}}}\right.}$ is equal to
$\Psi^{*}\hat{\omega}$ in a neighbourhood of the boundary of $B_{0}$. If this
is true, $\Psi$ acts by homotheties in a neighbourhood of $S$. Then the
restriction of $\hat{\omega}$ to $\hat{B}\backslash B_{0}=M_{i}$ can be
extended to a ${\mathbb{Z}}$-automorphic Kähler form on
$\tilde{M}=\bigcup_{i\in{\mathbb{Z}}}M_{i}$.

This is the strategy we follow, except that we replace $B$ by another strictly
holomorphically convex domain as in 6.

Step 2: Using the local $dd^{c}$-lemma, we can find a smooth function
$\varphi$ on $B_{0}$ such that
$dd^{c}\varphi=\hat{\omega}{\left|{}_{{\phantom{|}\\!\\!}_{B_{0}}}\right.}$.
Adding an appropriate affine function, we may assume that $\varphi$ reaches
its minimum in a point $x\in B_{0}$. Adding an appropriate constant, we can
assume that $U_{0}:=\varphi^{-1}(-\infty,0)$ has compact closure with smooth
boundary $B_{0}$. Let $U$ be the closure of $\Psi^{-1}(U_{0})$, and $\hat{U}$
the preimage of $U$ under the bimeromorphic contraction
$\hat{B}{\>\longrightarrow\>}B$. We obtain $M$ by gluing two boundary
components of $\hat{U}\backslash U_{0}$ as in 6.

Step 3: The map $\pi:\;\hat{U}{\>\longrightarrow\>}U$ is proper map of
manifolds of the same dimension. Therefore, the pushforward of a positive
$(p,p)$-form is a positive $(p,p)$-current.

Let $\pi_{*}\hat{\omega}$ be the pushforward of $\hat{\omega}$, considered as
a current on $U$. Using the $dd^{c}$-lemma for currents, we obtain
$\pi_{*}\hat{\omega}=dd^{c}f$, where $f$ is a plurisubharmonic function on $U$
which is smooth outside of the singularities of $\pi$.

Denote by $R$ the boundary of $U\subset{\mathbb{C}}^{2}$. Then $f$ is smooth
and strictly plurisubharmonic in a neighbourhood of $R$. Another
plurisubharmonic function in a neighbourhood of $R$ is obtained by taking
$f_{1}:=(\Psi^{-1})^{*}\varphi$, where $\varphi$ is the Kähler potential on
$U_{0}$ constructed in Step 2.

Rescaling $f$ if necessary and adding a constant, we may assume that
$-\varepsilon<f{\left|{}_{{\phantom{|}\\!\\!}_{R}}\right.}<0$ and
$|df|{\left|{}_{{\phantom{|}\\!\\!}_{R}}\right.}\ll\varepsilon$. Let $A$ be a
sufficienly big positive number, and $0<\delta\ll\varepsilon$. Then the
regularized maximum $\max_{\delta}$ of $f$ and $Af_{1}$ is equal to $Af_{1}$
in a very small neighbourhood of $R$ (because $f$ is negative on $R$ and
$f_{1}=0$ on $R$), and equal to $f$ in a neighbourhood $V$ of
$R_{\varepsilon}:=Af_{1}^{-1}(-2\varepsilon)$ because $|df|\ll A|df_{1}|$ and
as $Af_{1}$ goes to $-2\varepsilon$, $f$ does not go much below
$-\varepsilon$.

Replacing $\hat{\omega}$ by $dd^{c}\max_{\delta}(f,f_{1})$ on the annulus
between $R$ and $R_{2\varepsilon}$, we obtain a Kähler form
$\hat{\omega}_{1}$. Since $\max_{\delta}(f,f_{1})=f_{1}$ in a neighbourhood of
$R$, the map
$\Psi:\;(U,\hat{\omega}_{1}){\>\longrightarrow\>}(U_{0},\hat{\omega}_{1})$
acts by homothety mapping a neighbourhood of $R$ with the metric
$\hat{\omega}_{1}$ isometrically to a neighbourhood of $\Psi(R)$ with the
metric $A\hat{\omega}_{1}$ (however, $A\hat{\omega}_{1}=A\hat{\omega}$ outside
the annulus bounded by $R$ and $R_{2\varepsilon}$). We have constructed an LCK
metric on any Kato surface.

Remark 6.6: Prior to publishing his general result in [Bru2], M. Brunella
constructed examples of LCK metrics on Enoki surfaces ([Bru1]), which are
special cases of class VII surfaces admitting a global spherical shell. Even
before, LCK metrics on some Kato surfaces (hyperbolic Inoue and parabolic
Inoue) also appeared as anti-self-dual bihermitian metrics, obtained by
twistor methods in the work of A. Fujiki and M. Pontecorvo, see [FP1, FP2],
and [FP3] where the possibilities for the Lee class (cohomology class of the
Lee form $\theta\in\Lambda^{1}(M)$) of these structures is discussed (see also
[AD]). As M. Pontecorvo pointed out, [Po], it is implicitly shown there that
these Lee classes are different from the ones constructed by Brunella.

Acknowledgments: We are grateful to Georges Dloussky and Matei Toma for a
careful reading of an early version of this paper and to Eduardo Esteves for
an interesting discussion about elliptic surfaces.

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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 Str. 010702-Bucharest, Romania
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
also:
Laboratory of Algebraic Geometry,
Faculty of Mathematics, National Research University Higher School of
Economics, 7 Vavilova Str. Moscow, Russia
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Victor Vuletescu
University of Bucharest, Faculty of Mathematics,
14 Academiei str., 70109 Bucharest, Romania
<EMAIL_ADDRESS>