Algebraic dimension of complex nilmanifolds

Anna Fino, Gueo Grantcharov, Misha Verbitsky111Anna Fino is partially
supported by PRIN, FIRB and GNSAGA (Indam), Gueo Grantcharov is supported by a
grant from the Simons Foundation (#246184), and Misha Verbitsky is partially
supported by RSCF grant 14-21-00053 within AG Laboratory NRU-HSE.

Abstract

Let $M$ be a complex nilmanifold, that is, a compact quotient of a nilpotent
Lie group endowed with an invariant complex structure by a discrete lattice. A
holomorphic differential on $M$ is a closed, holomorphic 1-form. We show that
$a(M)\leqslant k$, where $a(M)$ is the algebraic dimension $a(M)$ (i.e. the
transcendence degree of the field of meromorphic functions) and $k$ is the
dimension of the space of holomorphic differentials. We prove a similar result
about meromorphic maps to Kahler manifolds.

###### Contents

  1. 1 Introduction
    1. 1.1 Nilmanifolds: definition and basic properties
    2. 1.2 Algebraic dimension and holomorphic differentials
  2. 2 The averaging formalism
  3. 3 Positive currents on nilmanifolds
    1. 3.1 Holomorphic differentials
    2. 3.2 Positive (1,1)-forms on a Lie algebra
    3. 3.3 Averaging the positive currents
  4. 4 Examples
    1. 4.1 Algebraic dimension of complex 2-tori
    2. 4.2 3-dimensional complex nilmanifolds

##  1 Introduction

###  1.1 Nilmanifolds: definition and basic properties

A nilmanifold is a compact manifold equipped with a transitive action of a
nilpotent Lie group. As shown by Malcev ([M]), every nilmanifold can be
obtained as a quotient of a nilpotent Lie group $G$ by a discrete lattice
$\Gamma$. Moreover, the group $G$ can be obtained as so-called Malcev
completion of $\Gamma$, that is, as a product of exponents of formal
logarithms of elements $\Gamma$. Therefore, any nilmanifold is uniquely
determined by its fundamental group, which is a discrete nilpotent torsion-
free group, and any such group uniquely determines a nilmanifold.

Since the work of Thurston ([T]), geometric structures on nilmanifolds are
used to provide many interesting examples (and counterexamples) in complex and
symplectic geometry. It was Thurston who realized that the Kodaira surface
(also known as a Kodaira-Thurston surface) is symplectic, but does not admit
any Kähler structure. In this way Thurston obtained a counterexample to a
result stated by H. Guggenheimer ([G]) in 1951. Guggenheimer claimed that the
Hodge decomposition is true for compact symplectic manifolds, but for
symplectic nilmanifolds this is usually false.

Before 1990-ies, a “complex nilmanifold” meant a compact quotient of a complex
nilpotent Lie group by a discrete, co-compact subgroup. The first non-trivial
example is given by so-called Iwasawa manifold ([FG]) which is obtained as a
quotient of the 3-dimensonal Lie group of upper triangular 3 by 3 matrices by
a discrete co-compact subgroup, for example the group of upper triangular
matrices with coefficients in ${\mathbb{Z}}[\sqrt{-1}\>]$.

Starting from late 1980-ies, a “complex nilmanifold” means a quotient of a
real nilpotent Lie group equipped with a left-invariant complex structure by
the left action of a discrete, co-compact subgroup ([CFG2]). This is the
notion we are going to use in this paper. This definition is much more
general, indeed, left-invariant complex structures are found on many even-
dimensional nilpotent Lie groups which are not complex. The complex structure
on a Kodaira surface is one of such examples.

Complex structures on a nilmanifold have a very neat algebraic
characterization. Let $G$ be a real nilpotent Lie group, and ${\mathfrak{g}}$
is Lie algebra. By Newlander-Nirenberg theorem, a complex structure on $G$ is
the same as a sub-bundle $T^{1,0}G\subset TG\otimes_{\mathbb{R}}{\mathbb{C}}$
such that $[T^{1,0}G,T^{1,0}G]\subset T^{1,0}G$ and
$T^{1,0}G\oplus\overline{T^{1,0}G}=TG\otimes_{\mathbb{R}}{\mathbb{C}}$. The
left-invariant sub-bundles in $T^{1,0}G$ are the same as subspaces
$W\subset{\mathfrak{g}}\otimes_{\mathbb{R}}{\mathbb{C}}$, and the condition
$[T^{1,0}G,T^{1,0}G]\subset T^{1,0}G$ is equivalent to $[W,W]\subset W$.
Therefore, left-invariant complex structures on $G$ are the same as complex
sub-algebras
${\mathfrak{g}}^{1,0}\subset{\mathfrak{g}}\otimes_{\mathbb{R}}{\mathbb{C}}$
satisfying
${\mathfrak{g}}^{1,0}\oplus\overline{{\mathfrak{g}}^{1,0}}={\mathfrak{g}}\otimes_{\mathbb{R}}{\mathbb{C}}$.

A real nilmanifold is obtained as an iterated fibration with fibers which are
compact tori. It is natural to ask if any complex nilmanifold can be obtained
as an iterated fibration with fibers which are complex tori. The answer is
negative: see e.g. [R].

However, a weaker statement is still true. If we replace fibrations of
nilmanifolds by homomorphisms of their Lie algebras, it is possible to
construct a homomorphism
$\psi:\;{\mathfrak{g}}{\>\longrightarrow\>}{\mathfrak{a}}$ to a complex
abelian Lie algebra compatible with a complex structure. Since
${\mathfrak{a}}$ is abelian, $\ker\psi$ necessarily contains the commutator
$[{\mathfrak{g}},{\mathfrak{g}}]$. Since it is complex, $\ker\psi$ contains
$[{\mathfrak{g}},{\mathfrak{g}}]+I[{\mathfrak{g}},{\mathfrak{g}}]$.

The quotient algebra
${\mathfrak{g}}/[{\mathfrak{g}},{\mathfrak{g}}]+I[{\mathfrak{g}},{\mathfrak{g}}]$
is called the algebra of holomorphic differentials on $G$, denoted by
${\mathfrak{H}}^{1}(M)$. Its dimension is always positive ([S]).

In this paper, we study algebraic objects (such as meromorphic functions) on
complex nilmanifolds. We prove the following theorem.

Theorem 1.1:  Let $M=G/\Gamma$ be a complex nilmanifold, and $\Sigma$ be a a
foliation obtained by left translates of
$[{\mathfrak{g}},{\mathfrak{g}}]+I[{\mathfrak{g}},{\mathfrak{g}}]$, where $I$
is the complex structure operator, and ${\mathfrak{g}}=T_{e}G$ the Lie algebra
of $G$. Then all meromorphic functions on $M$ are constant on the leaves of
$\Sigma$.

Proof: See Subsection 3.3.

###  1.2 Algebraic dimension and holomorphic differentials

Recall that a positive closed (1,1)-current $T$ on a complex manifold is said
to have analytic singularities (see [Bouck]) if locally
$T=\theta+dd^{c}\varphi$ for a smooth form $\theta$ and a plurisubharmonic
function $\varphi=c\log(|f_{1}|^{2}+...+|f_{n}|^{2})$ where $f_{1},...f_{n}$
are analytic functions and $c$ a constant. Such currents have decomposition
into absolutely continuous and singular part, where the absolutely continuous
part is positive and closed.

Definition 1.2: Let $M$ be a complex manifold. The Kähler rank $k(M)$ of $M$
is the maximal rank of the absolutely continuous part of a positive, closed
(1,1)-current on $M$ with analytic singularities.

Definition 1.3: The algebraic dimension $a(M)$ of a complex manifold is the
transcendence degree of its field of meromorphic functions.

Let $X$ be a complex manifold, and
$\varphi:\;X\dashedrightarrow{\mathbb{C}}^{n}$ a meromorphic map defined by
generators of the field of meromorphic functions. An algebraic reduction of
$X$ ([Ca81], [Ue75]) is a compactification of $\varphi(X)$ in
${\mathbb{C}}P^{n}\supset{\mathbb{C}}^{n}$. It is known to be a compact,
algebraic variety ([Ca81], [Ue75]).

We should note that the map $\varphi$ is defined for more general spaces $X$.
For smooth manifolds we’ll use the following [Pet, Definition-Theorem 6.5].

Definition 1.4:  Let $M$ be a compact complex manifold. Then there exists a
smooth projective manifold $X$, a rational map $\varphi:\;M\dashedrightarrow
X$ and a diagram where $X^{\prime}$ is smooth and the top two arrows are
proper holomorphic maps with $a$ a proper bimeromorphic modification, such
that the corresponding fields of meromorphic functions coincide:
$\operatorname{Mer}(M)=\operatorname{Mer}(X)$. We call the map
$\varphi:\;M\dashedrightarrow X$ algebraic reduction of $M$.

Definition 1.5:  Let $\eta$ be a positive $(1,1)$-form on $X$. The current
$T_{\eta}$ is defined as $a_{*}b^{*}\eta$; since pushforward of a form is a
current, it is a current, and not a form. Since $a$ is one-to-one everywhere,
except on an analytic set $E\subset X^{\prime}$, the current $a_{*}b^{*}\eta$
is smooth outside of $E$. Note also that the positivity and closedness are
preserved, as well as the rank in a general point. We call $T_{\eta}$ the
current induced by $\eta$ on $M$. It is easy to check that $T_{\eta}$ has
analytic singularities if $\eta$ is closed and positive.

Claim 1.6: Let $M$ be a complex variety. Then the algebraic dimension is
bounded by the Kähler rank:

$a(M)\leqslant k(M).$ (1.1)

Proof: Let $\varphi:\;M\dashedrightarrow X$ be the algebraic reduction map.
Pullback of a Kähler form from $X$ to $M$ is a current of rank $\dim X$ at all
points where it is absolutely continuous.

We are going to estimate $a(M)$ in terms of holomorphic differentials on $M$.

Definition 1.7: A holomorphic differential on a compact complex manifold is a
closed, holomorphic 1-form.

Theorem 1.8:  Let $M$ be a complex nilmanifold, ${\mathfrak{H}}^{1}(M)$ the
space of holomorphic differentials on $M$, and $a(M)$ its algebraic dimension.
Then

$a(M)\leqslant\dim{\mathfrak{H}}^{1}(M).$

Proof: See Subsection 3.3.

Remark 1.9: The same estimate holds for complex parallelizable manifolds; see
[W, Proposition 3.16.1]. Moreover, 1.1 is also true in this situation: all
meromorphic functions are constant on the fibers of the Albanese map.

This result is implied by 1.2 and the following expression of the Kähler rank.

Theorem 1.10:  Let $M$ be a complex nilmanifold, ${\mathfrak{H}}^{1}(M)$ the
space holomorphic differentials on $M$, and $k(M)$ its Kähler rank. Then
$k(M)=\dim{\mathfrak{H}}^{1}(M)$.

Proof: Consider the projection
${\mathfrak{g}}{\>\longrightarrow\>}{\mathfrak{a}}$, where
${\mathfrak{a}}=\frac{{\mathfrak{g}}}{[{\mathfrak{g}},{\mathfrak{g}}]+I([{\mathfrak{g}},{\mathfrak{g}}])}$.
Since $[{\mathfrak{g}},{\mathfrak{g}}]+I([{\mathfrak{g}},{\mathfrak{g}}])$ is
$I$-invariant, $a$ has a complex structure and this map is compatible with it.
Consider the Chevalley differential $d$ on the Lie algebras of
${\mathfrak{g}}$ and ${\mathfrak{a}}$. Since ${\mathfrak{a}}$ is an abelian
algebra, any 2-form on ${\mathfrak{a}}$ is closed (and gives a closed 2-form
on the corresponding Lie group). Taking a positive definite Hermitian form, we
obtain a positive current of rank $\dim{\mathfrak{a}}={\mathfrak{H}}^{1}(M)$
on $M$. There are no currents with greater rank by 3.3.

The same argument implies the following useful corollary.

Corollary 1.11:  Let $M$ be a complex nilmanifold, ${\mathfrak{g}}$ the Lie
algebra of the corresponding Lie group, and
${\mathfrak{h}}:=[{\mathfrak{g}},{\mathfrak{g}}]+I([{\mathfrak{g}},{\mathfrak{g}}])$
the algebra constructed as above. Denote by ${\mathfrak{h}}_{1}$ a smallest
$I$-invariant rational subspace of ${\mathfrak{g}}$ containing
${\mathfrak{h}}$. Let $T$ be a complex torus obtained as quotient of
${\mathfrak{g}}/{\mathfrak{h}}_{1}$ by its integer lattice. Consider the
natural holomorphic projection $\Psi:\;M{\>\longrightarrow\>}T$. Then any
meromorphic map to a Kähler manifold is factorized through $\Psi$.

Proof: Let $\psi:\;M\dashedrightarrow X$ be a meromorphic map to a Kähler
manifold $(X,\omega)$. For general $x\in X$, the zero space of the positive
closed current $\psi^{*}\omega$ contains ${\mathfrak{h}}$, hence the fibers
$F_{x}:=\psi^{-1}(x)$ are tangent to ${\mathfrak{h}}$. The smallest compact
complex subvariety of $M$ containing a leaf of the foliation associated with
${\mathfrak{h}}$ is the corresponding leaf of ${\mathfrak{h}}_{1}$. Passing to
the closures of the leaves of ${\mathfrak{h}}$, we obtain that $F_{x}$ contain
leaves of ${\mathfrak{h}}_{1}$. However, $T$ is the leaf space of
${\mathfrak{h}}_{1}$.

Remark 1.12:  For a general compact complex manifold $X$, Albanese variety
$\operatorname{Alb}(X)$ is defined as the quotient of the dual space of the
space of holomorphic differentials $H^{0}(X,d\mathcal{O})^{*}$ by the minimal
closed complex subgroup containing the image of $H^{1}(X,\mathbb{Z})$ under
the map

$H^{1}(X,\mathbb{Z})\rightarrow H^{1}(X,\mathbb{C})\rightarrow
H^{0}(X,d\mathcal{O})^{*}$

(see [R]). The Albanese map
$\operatorname{Alb}:X\rightarrow\operatorname{Alb}(X)$ is give n by
integration along paths starting at a fixed point. It has the functorial
property tha t any map from $X$ onto tori is factored through the Albanese map
([Ue75]). In [R] (section 2.1) the Albanese variety for a complex nilma nifold
$M$ is described in terms of the space $\mathfrak{h_{1}}$ as

$\operatorname{Alb}(M)=\frac{H^{0}(X,d\cal{O})^{*}/p(\mathfrak{h_{1}})}{\operatorname{im}(H^{1}(X,\mathbb{Z})\rightarrow
H^{0}(X,d\mathcal{O}))^{*}/p(\mathfrak{h_{1}})}=T,$

where $\mathfrak{h_{1}}$ is the same as in 1.2. Then we obtain that
$T=\operatorname{Alb}(M)$ and

$a(M)=a(\operatorname{Alb}(M)).$

##  2 The averaging formalism

Let $M=\Gamma\backslash G$ be a compact nilmanifold and $\nu$ a volume element
on $M$ induced by a the Haar measure on the Lie group $G$ [Mi]. After a
rescaling, we can suppose that $M$ has volume $1$. Notice that the Haar
measure on $G$ is bi-invariant, because $G$ admits a lattice, and any Lie
group admitting a lattice is unimodular.

Given any covariant $k$-tensor field $T:TM\times\ldots\times
TM\rightarrow{\mathcal{C}}^{\infty}(M)$ on the nilmanifold $M$, one can define
a covariant $k$-tensor

$T_{inv}:\mathfrak{g}\times\ldots\times\mathfrak{g}\rightarrow{\mathbb{R}}$

on the Lie algebra $\mathfrak{g}$ of $G$ by

$T_{inv}(x_{1},\ldots,x_{k})=\int_{p\in
M}T_{p}(x_{1}|_{p},\ldots,x_{k}|_{p})\nu,$

for every $x_{1},\ldots,x_{k}\in\mathfrak{g}$, where $x_{l}|_{p}$ is the
restriction of the left-invariant vector field $X_{l}$ to $p$. Clearly,
$T_{inv}=T$ for any tensor field T coming from a left-invariant one. In [B] it
is shown that that if $\alpha$ is a differential $k$-form on $M$, then
$(d\alpha)_{inv}=d(\alpha_{inv})$. Moreover,
$(\alpha_{inv}\wedge\beta)_{inv}=\alpha_{inv}\wedge\beta_{inv}$, for every
differential forms $\alpha$ and $\beta$ on $M$.

We call the map $\operatorname{\sf Av}:\;(T^{*})^{\otimes
k}{\>\longrightarrow\>}({\mathfrak{g}}^{*})^{\otimes k}$, $\operatorname{\sf
Av}(T):=T_{inv}$ averaging on a nilmanifold. The averaging defines a linear
map $\tilde{\nu}:\Omega^{k}(M)\rightarrow\Lambda^{k}\mathfrak{g}^{*}$, given
by $\tilde{\nu}(\alpha)=\alpha_{inv}$ for every $k$-form
$\alpha\in\Omega^{k}(M),$ which commutes with the differentials.

Moreover, by Nomizu theorem [N] $\tilde{\nu}$ induces an isomorphism
$H^{k}(M)\rightarrow H^{k}(\mathfrak{g})$ between the kth cohomology groups
for every $k$. In particular, every closed $k$-form $\alpha$ on M is
cohomologous to the invariant $k$-form $\alpha_{inv}$ obtained by the
averaging (see also [Ug]) Indeed, by Nomizu Theorem $\alpha=\beta+d\gamma$,
with $\beta$ invariant closed $k$-form. By using the averaging we have
$\alpha_{inv}=\beta+d\gamma_{inv}$ and so $\alpha$ is cohomologous to
$\alpha_{inv}$.

##  3 Positive currents on nilmanifolds

###  3.1 Holomorphic differentials

Recall that holomorphic differentials on a complex manifold are closed,
holomorphic 1-forms.

Definition 3.1: Let $M=\Gamma\backslash G$ be a nilmanifold. A differential
form on $M$ is called invariant if its pullback to $M$ is invariant with
respect to the left action of $G$ on itself.

Remark 3.2: Let $M=\Gamma\backslash G$ be a nilmanifold, ${\mathfrak{g}}$ the
Lie algebra of $G$. Clearly, invariant differential forms are identified with
$\Lambda^{*}({\mathfrak{g}})$. Moreover, they are preserved by de Rham
differential, which is identified with the Chevalley differential on
$\Lambda^{*}({\mathfrak{g}})$.

Proposition 3.3: Let $(M,I)$ be a complex nilmanifold, and $h$ a holomorphic
differential. Then $h$ is an invariant differential form.

Proof: Let $\nu$ be a volume element on $M$ induced by a bi-invariant one on
the Lie group $G$ such that $M$ has volume equal to $1$. A holomorphic
differential $h$ is cohomologous to the invariant form $h_{inv}$ obtained by
the averaging process. Since $I$ is invariant, $h_{inv}$ has to be of type
$(1,0)$ and thus $h=h_{inv}.$ Indeed, closed (1,0)-forms cannot be exact,
because they are holomorphic, hence (if exact) equal to differentials of a
global holomorphic function.

Corollary 3.4:  Let $M=\Gamma\backslash G$ be a complex nilmanifold, and
${\mathfrak{g}}$ its Lie algebra, and ${\mathfrak{H}}^{1}(M)$ the space of
holomorphic differentials. Then

${\mathfrak{H}}^{1}(M)=\left(\frac{{\mathfrak{g}}\otimes{\mathbb{C}}}{{\mathfrak{g}}^{1}+I({\mathfrak{g}}^{1})}\right)^{*},$

where ${\mathfrak{g}}^{1}=[{\mathfrak{g}},{\mathfrak{g}}]$ denotes the
commutator of ${\mathfrak{g}}$.

Proof: Let $h$ be a holomorphic differential. Since $h$ is invariant then it
can be identified with an element of
$({\mathfrak{g}}\otimes{\mathbb{C}})^{*}$. Moreover, $h=\alpha+iI\alpha$, with
$\alpha\in{\mathfrak{g}}^{*}$, $d\alpha=0$ and $d(I\alpha)=0$. By the
conditions

$d\alpha(x,y)=-\alpha([x,y])=0,\quad d(I\alpha)(x,y)=\alpha(I[x,y])=0,$

for every $x,y\in{\mathfrak{g}}$, we get
$\alpha({\mathfrak{g}}^{1})=\alpha(I{\mathfrak{g}}^{1})=0$.

###  3.2 Positive (1,1)-forms on a Lie algebra

Throughout this subsection, we fix a nilpotent Lie algebra ${\mathfrak{g}}$
with a complex structure $I\in\operatorname{End}({\mathfrak{g}})$ satisfying
the integrability condition

$[{\mathfrak{g}}^{1,0},{\mathfrak{g}}^{1,0}]\subset{\mathfrak{g}}^{1,0}.$

Definition 3.5: A semipositive Hermitian form on $({\mathfrak{g}},I)$ is a
real form $\eta\in\Lambda^{2}({\mathfrak{g}}^{*})$ which is $I$-invariant
(that is, of Hodge type (1,1)) and satisfies $\eta(x,Ix)\geqslant 0$ for each
$x\in{\mathfrak{g}}$. It is called positive definite Hermitian if this
inequality is strict for all $x\neq 0$.

Definition 3.6: A subalgebra ${\mathfrak{a}}\subset{\mathfrak{g}}$ is called
holomorphic if $I({\mathfrak{a}})={\mathfrak{a}}$ and
$[{\mathfrak{g}}^{0,1},{\mathfrak{a}}^{1,0}]^{1,0}\subset{\mathfrak{a}}^{1,0}$.

Claim 3.7: Let ${\mathfrak{a}}\subset{\mathfrak{g}}$ be a vector subspace, and
$B:={\mathfrak{a}}\cdot G$ the corresponding left-invariant sub-bundle in
$TG$. Then

  * •

$B$ is involutive (that is, Frobenius integrable) iff ${\mathfrak{a}}$ is a
Lie subalgebra of ${\mathfrak{g}}$.

  * •

$B$ is a holomorphic sub-bundle iff ${\mathfrak{a}}$ is a holomorphic
subalgebra.

Proof: Let $x,y\in{\mathfrak{a}}$ and denote by the same letters the
corresponding left-invariant vector fields. Clearly, $B$ is involutive if and
only if ${\mathfrak{a}}$ is a Lie subalgebra of ${\mathfrak{g}}$. Similarly we
have that $B$ is holomorphic if $[x+iIx,y-iIy]\in{\mathfrak{a}}^{1,0}$, for
every $x\in\mathfrak{g}$ and $y\in\mathfrak{a}.$

Remark 3.8: Note that $V={\mathfrak{g}}^{(1,0)}+{\mathfrak{a}}^{(0,1)}$ is
involutive iff ${\mathfrak{a}}$ is holomorphic and
$V+\overline{V}={\mathfrak{g}}^{c}$. So $V$ is an “elliptic structure” in the
terminology of [Jac], so by [Jac] it defines a holomorphic foliation.

We also note the obvious

Claim 3.9:  If $V_{1}$ and $V_{2}$ are two elliptic structures in terminology
of [Jac] on a complex manifold, containing the $(1,0)$ tangent bundle, then
$V_{1}\cap V_{2}$ is also an elliptic structure.

Definition 3.10: Let $\eta$ be a semipositive Hermitian form on
$({\mathfrak{g}},I)$, and $N(\eta)$ the subspace in ${\mathfrak{g}}$
consisting of all vectors $x$ such that $\eta(x,Ix)=0$. Then $N(\eta)$ is
called the null-space of $\eta$.

In general we have the following

Claim 3.11: The nullspace

$N=\\{x\in{\mathfrak{g}}\,\mid\iota_{x}\eta=0\\}$

of a closed form $\eta\in\Lambda^{r}\mathfrak{g}^{*}$ is a Lie subalgebra of
$\mathfrak{g}$.

Proof: Take $x,y\in N$ and arbitrary vectors
$z_{1},\ldots,z_{r-1}\in\mathfrak{g}$. Then, by Cartan’s formula,
$d\eta(x,y,z_{1},\ldots,z_{r-1})=\eta([x,y],z_{1},\ldots z_{r-1})=0$, since
the rest of the terms vanish, because $x,y\in N$. Therefore
$\eta([x,y],z_{1},\ldots z_{r-1})=0$ for any
$z_{1},\ldots,z_{r-1}\in\mathfrak{g}$, this means that $\iota_{[x,y]}\eta=0$,
i.e $[x,y]\in N$.

Theorem 3.12:  Let $\eta$ be a semipositive Hermitian form on
$({\mathfrak{g}},I)$. Assume that its nullspace $N(\eta)$ is a holomorphic
subalgebra. Then $N(\eta)$ contains ${\mathfrak{g}}^{1}+I{\mathfrak{g}}^{1}$,
where ${\mathfrak{g}}^{1}=[{\mathfrak{g}},{\mathfrak{g}}]$.

Proof: When the cohomology class of $\eta$ is rational, as happens in most
applications, 3.2 has a simple proof. Since $[\eta]$ is rational, it can be
represented by a rational form
$\eta_{\mathbb{Q}}\in\Lambda^{2}({\mathfrak{g}})$. Therefore, the leaves of
$N(\eta)$ are rational Lie subalgebras in ${\mathfrak{g}}$. By Malcev’s
theorem, the leaves of $N(\eta)$ are compact. By construction, the leaf space
$X$ of $N(\eta)$ is equipped with a transitive action by a nilpotent Lie
group, hence it is a nilmanifold. Finally, $X$ inherits the complex structure
from $X$, and $\eta$ defines a Kähler metric on $X$. However, a nilmanifold
can be Kähler only if its fundamental group is abelian ([BG]). Therefore,
$N(\eta_{\mathbb{Q}})$ contains $[{\mathfrak{g}},{\mathfrak{g}}]$.

For general $\eta$, 3.2 has a different (more complicated) proof.

Since $N(\eta)=\mathfrak{a}$ is holomorphic, we have

$[y+iIy,x-iIx]^{1,0}\in{\mathfrak{a}}^{1,0},$ (3.1)

for every $x\in\mathfrak{a}$ and for every $y\in\mathfrak{g}$. By a direct
computation we obtain

$\begin{array}[]{lcl}[y+iIy,x-iIx]^{1,0}&=&([y,x]+[Iy,Ix]+I[Iy,x]-I[y,Ix])\\\\[3.0pt]
&&-iI([y,x]+[Iy,Ix]+I[Iy,x]-I[y,Ix]).\end{array}$

Therefore, by the condition (3.1) we get

$[y,x]+[Iy,Ix]+I[Iy,x]-I[y,Ix]\in\mathfrak{a},\quad\forall
x\in\mathfrak{a},\,\forall y\in\mathfrak{g}.$ (3.2)

By using the integrability condition

$[Iy,Ix]=[y,x]+I[Iy,x]+I[y,Ix]$

we have

$I[y,Ix]=[Iy,Ix]-[y,x]-I[Iy,x]$

and therefore the condition (3.2) becomes

$2([y,x]+I[Iy,x])\in\mathfrak{a},\quad\forall x\in\mathfrak{a},\,\forall
y\in\mathfrak{g},$

i.e.

$\eta([y,x],z)=-\eta(I[Iy,x],z),\quad\forall x\in\mathfrak{a},\,\forall
y,z\in\mathfrak{g}.$

Therefore

$\eta([y,x],I[y,x])=-\eta(I[Iy,x],I[y,x])=-\eta([x,Iy],[x,y]).$

By $d\eta=0$, one gets

$\eta([x,y],w)=\eta([x,w],y),$

for every $x\in\mathfrak{a},y,z\in\mathfrak{g}.$ Thus

$\eta([x,Iy],[x,y])=\eta(ad_{x}^{2}(y),Iy),\quad\forall
x\in\mathfrak{a},\,\forall y\in\mathfrak{g}$

and consequently

$\eta([y,x],I[y,x])=-\eta(ad_{x}^{2}(y),Iy),\quad\forall
x\in\mathfrak{a},\,\forall y\in\mathfrak{g}.$ (3.3)

By using (3.3), it is possible to show that $\mathfrak{a}$ is an ideal of
$\mathfrak{g}$, i.e. that $[y,x]\in\mathfrak{a}$, for every $x\in\mathfrak{a}$
and $y\in\mathfrak{g}$.

Since $\eta$ is a semipositive (1,1)-form and $\mathfrak{a}$ is its null-
space, the relation $\eta([y,x],I[y,x])=0$ implies that
$[x,y]\in\mathfrak{a}$. Therefore, by (3.3), in order to prove that
$[y,x]\in\mathfrak{a}$, for every $x\in\mathfrak{a}$ and for every
$y\in\mathfrak{g}$, it is sufficient to show that $[x,[x,y]]\in\mathfrak{a}$
for any $x\in\mathfrak{a}$. This would follow if we prove that

$[\mathfrak{a},{\mathfrak{g}}^{1}]\subset\mathfrak{a}.$ (3.4)

Since $\mathfrak{g}$ is nilpotent there exists $s$ such that
$\mathfrak{g}^{s}=\\{0\\}$ and $\mathfrak{g}^{s-1}\neq\\{0\\}$ and we have the
descending series of ideals

$\mathfrak{g}={\mathfrak{g}}^{0}\supset{\mathfrak{g}}^{1}\supset\ldots\supset{\mathfrak{g}}^{i}\supset{\mathfrak{g}}^{i+1}\supset\ldots\supset{\mathfrak{g}}^{s-1}\supset{\mathfrak{g}}^{s}=\\{0\\}.$

Now we can prove that $[\mathfrak{a},{\mathfrak{g}}^{1}]\subset\mathfrak{a}$
by induction on $i$ in the following way: by using (3.3) we can show that

(A) if the condition $[\mathfrak{a},{\mathfrak{g}}^{i+1}]\subset\mathfrak{a}$
holds, then the condition
$[\mathfrak{a},{\mathfrak{g}}^{i}]\subset\mathfrak{a}$ holds.

Since $\mathfrak{g}$ is nilpotent there exists $s$ such that
$\mathfrak{g}^{s}=\\{0\\}$ and $\mathfrak{g}^{s-1}\neq\\{0\\}$.

At the first step $i={s-1}$ we have that (A) holds. So by induction we obtain
that (A) holds for $i=0$. Consequently, $\mathfrak{a}=N(\eta)$ is an ideal of
$\mathfrak{g}$ and $\eta$ induces a Kähler form on the nilpotent Lie algebra
${\mathfrak{g}}/{\mathfrak{a}}$. By [BG], the Kähler nilpotent Lie algebra
${\mathfrak{g}}/{\mathfrak{a}}$ has to be abelian. Therefore
${\mathfrak{g}}^{1}\subset\mathfrak{a}$. Since $\eta$ is (1,1)-form, its null-
space $\mathfrak{a}$ is $I$-invariant, hence $\mathfrak{a}$ contains
$\mathfrak{g}^{1}+I\mathfrak{g}^{1}$.

###  3.3 Averaging the positive currents

The following result directly follows from the averaging.

Proposition 3.13:  Let $M=\Gamma\backslash G$ be a compact quotient of a
unimodular Lie group $G$ by a lattice $\Gamma$ and $I$ the complex structure
on $M$ obtained from an invariant complex structure on $G$. Let $T_{\eta}$ be
the positive, closed (1,1)-current induced by the algebraic reduction
$\varphi:M\rightarrow X$ from some Kähler form $\eta$ on $X$ (1.2). If
$\operatorname{\sf Av}(T)$ is its average, then $\operatorname{\sf Av}(T)$ is
a semipositive, closed, $G$-invariant differential form, and its rank is no
less than the rank of the absolutely continuous part of $T_{\eta}$.

Proof: If $X$ and $Y$ are left-invariant vector fields on $M$, then
$T_{\eta}(X,Y)$ is a measurable function when we consider $T_{\eta}$ as a form
with distributional coefficients in local coordinates. So $\operatorname{\sf
Av}(T)$ is well defined as in Section 2. Then $\operatorname{\sf Av}(T)$ is a
closed invariant form of type (1,1) and the only thing to check is the
statement about its rank. By the definition it follows that $\operatorname{\sf
Av}(T)(X,IX)=0\Leftrightarrow T_{p}(X|_{p},IX|_{p})=0$ for almost all $p\in
M$. So $X$ is in the kernel of $\operatorname{\sf Av}(T)$ only if it is in the
kernel of $T_{p}$ for almost all $p$.

Remark 3.14: As a corollary we obtain that if such space admits a Kähler
current, it is Kähler. In particular from [DP] it follows that such spaces are
never in Fujiki’s class $\mathcal{C}$. Note that the proof of this fact in
[DP] uses also the Kähler current arising from the pull-back of a Kähler form.

Proposition 3.15:  Let $T$ be a positive, closed (1,1)-current on a
nilmanifold $M=G/\Gamma$, and ${\cal F}$ the null-space foliation of its
absolutely continuous part. Then the sub-bunlde associated with ${\cal F}$
contains a homogeneous sub-bundle $\Sigma$ obtained by left translates of
${\mathfrak{g}}^{1}+I{\mathfrak{g}}^{1}$, where
${\mathfrak{g}}^{1}=[{\mathfrak{g}},{\mathfrak{g}}]$, and ${\mathfrak{g}}$ is
the Lie algebra of $G$.

Proof: Let $\operatorname{\sf Av}$ be the averaging map defined in Section 2.
The nullspace of the form $\operatorname{\sf Av}(T)$ is contained in the
intersection of all left translates of ${\cal F}$, hence by 3.2 it is also
holomorphic. Then 3.2 implies that $N(\operatorname{\sf Av}(T))$ contains
${\mathfrak{g}}^{1}+I{\mathfrak{g}}^{1}$.

Proof of 1.1: Let $M{\>\longrightarrow\>}X$ be the algebraic reduction map
(1.2), and $\eta$ the pullback of the Kähler form on $X$. Averaging (3.3)
transforms $\eta$ into an invariant, closed, semipositive form. Then $\eta$
vanishes on $\Sigma$ by 3.3.

Proof of 1.2: Now we can prove 1.2. Let $M=\Gamma\backslash G$ be a
nilmanifold, and $\varphi:\;M\dashedrightarrow X$ the algebraic reduction map.
The pullback $\varphi^{*}\omega_{X}$ of a Kähler form $\omega_{X}$ is a
current $T$ on $M$ (1.2). By 3.3, the rank of its absolutely continuous part
is no greater that

$\dim\frac{{\mathfrak{g}}}{{\mathfrak{g}}^{1}+I{\mathfrak{g}}^{1}}=\dim{\mathfrak{H}}^{1}(M)$

(the latter equality follows from 3.1).

##  4 Examples

All 2-dimensional compact complex nilmanifolds are classified and correspond
to tori and primary Kodaira surfaces. Their algebraic dimension is known. In
this section we’ll consider the algebraic dimension of the complex
nilmanifolds in dimension 3 and note that for other complex homogeneous spaces
the inequality $a(M)\leqslant\mathfrak{H}^{1}(M)$ may not hold.

Many nilmanifolds admit holomorphic fibrations and we’ll need the following:

Remark 4.1:  In general (see [Ue75, Theorem 3.8]), if a complex manifold $M$
is the total space of a holomorphic fibration $\pi:M\rightarrow B$ we always
have the inequality

$a(M)\geqslant a(B).$

###  4.1 Algebraic dimension of complex 2-tori

Following [BL] we have the following description of the algebraic dimension of
the complex 2-tori.

Let $T^{4}$ be the tori defined as ${\mathbb{R}}^{4}/{\mathbb{Z}}^{4}$ where
${\mathbb{Z}}^{4}$ is the standard lattice in ${\mathbb{R}}^{4}$. Let
$J\in\operatorname{End}({\mathbb{R}}^{4}),J=\left(\begin{array}[]{lcl}A&B\\\
C&D\end{array}\right)$ be a complex structure with $A,B,C,D$ $2\times
2$-blocks and $B$ nondegenerate. From [BL] (p.10) we can identify what is the
period lattice of the complex tori with structure $J$. If
$X={\mathbb{C}}^{2}/(\tau,Id_{2}){\mathbb{Z}}^{4}$ is a complex tori defined
by a complex $2\times 2$ matrix $\tau$, then the complex structure $J_{\tau}$
on $T^{4}$ such that $X\cong(T^{4},J_{\tau})$ as complex manifold is given by

$J_{\tau}=\left(\begin{array}[]{lcl}y^{-1}x&y^{-1}\\\
-y-xy^{-1}x&-xy^{-1}\end{array}\right),$

where $x$ and $y$ are the real and imaginary parts of $\tau$. Reversing the
construction gives that for $J$ as above,

$\tau_{J}=B^{-1}A+iB^{-1}.$

We also need the relation to complex structures, defined in terms of a basis
of $(1,0)$-forms. If $J_{0}$ is a fixed complex structure and
$\omega_{j}=e_{j}+\sqrt{-1}\>J_{0}e_{j},j=1,2$ is a basis of (linear)
$(1,0)$-forms for $J_{0}$, we define another complex structure $J$ as

$\begin{array}[]{lll}\alpha_{1}&=&\omega_{1}+a\overline{\omega_{1}}+b\overline{\omega_{2}}\\\
\alpha_{2}&=&\omega_{2}+c\overline{\omega_{1}}+d\overline{\omega_{2}}\end{array}$
(4.1)

being the basis of $(1,0)$-forms of $J$. If
$X=\left(\begin{array}[]{lcl}a&b\\\ c&d\end{array}\right)=X_{1}+iX_{2}$, then
the relation between $X$ and the matrix representing $J$ in the basis
$(e_{i},J_{0}e_{i})$ is given by

$J=\left(\begin{array}[]{lcl}Id+X_{1}&X_{2}\\\ X_{2}&Id-
X_{1}\end{array}\right)^{-1}\left(\begin{array}[]{lcl}0&Id\\\
-Id&0\end{array}\right)\left(\begin{array}[]{lcl}Id+X_{1}&X_{2}\\\ X_{2}&Id-
X_{1}\end{array}\right)$

We’ll use the explicit form of $J$ and $\tau_{J}$ when
$X_{1}=\left(\begin{array}[]{lcl}0&a\\\ 0&0\end{array}\right)$ and
$X_{2}=\left(\begin{array}[]{lcl}0&b\\\ 0&0\end{array}\right)$. Direct
calculation (using the fact that
$X_{1}^{2}=X_{2}^{2}=X_{1}X_{2}=X_{2}X_{1}=0)$ gives
$J=\left(\begin{array}[]{lcl}2X_{2}&Id-2X_{1}\\\
-Id-2X_{1}&-2X_{2}\end{array}\right)$ and

$\tau_{J}=\left(\begin{array}[]{lcl}i&2a+2bi\\\ 0&i\end{array}\right)=iID+2X.$

To determine the algebraic dimension of $(T^{4},J)$, we need first the Neron-
Severi group $NS(J)$ of $J$. Let $\tau_{ij}$ are the components of $\tau_{J}$
and

$E=\left(\begin{array}[]{llll}0&a&b&c\\\ -a&0&d&e\\\ -b&-d&0&f\\\
-c&-e&-f&0\end{array}\right)\in M_{4}({\mathbb{Z}})$

be an integral matrix. Then $NS(J)=\\{E\in
M_{4}(Z)|a+d\tau_{11}-b\tau_{12}+f\tau_{21}-c\tau_{22}+edet(\tau)=0\\}$. With
these notations in mind, the algebraic dimension of $(T^{4},J)$ is determined
by

$a(J)=\frac{1}{2}{\rm max}\\{\operatorname{\sf rank}(J^{T}E)|E\in
NS(J),J^{T}E\geqslant 0\\},$

where the superscript $T$ means transposition. Note that not all complex
structures are described in this way - we have the non-degeneracy condition on
$B$ which is required for $(\tau,Id_{2})$ to be a period matrix. It is well
known that $a(J)$ could be any of 0,1 or 2.

The integrality condition leads to the fact that generically $a(J)=0$. For
$a(J)\geqslant 0$ from [BL] p.59, we know that $a(J)=1$ exactly when the torus
admits a period matrix $(\tau,Id_{2})$ with

$\tau=\left(\begin{array}[]{lcl}\tau_{1}&\alpha\\\
0&\tau_{2}\end{array}\right)$

with

$\alpha\notin(\tau_{1},1)M_{2}(\mathbb{Q})\left(\begin{array}[]{l}1\\\
\tau_{2}\end{array}\right),$

where $M_{2}(\mathbb{Q})$ is the set of $2\times 2$-matrices. In particular in
(4.1), when $X=\left(\begin{array}[]{lcl}0&\sqrt{2}-i\sqrt{3}\\\
0&0\end{array}\right)$, the algebraic dimension of $(T^{4},J)$ is one.

###  4.2 3-dimensional complex nilmanifolds

Definition 4.2:  Let $\mathfrak{g}$ be a nilpotent Lie algebra. A rational
structure for $\mathfrak{g}$ is a subalgebra $\mathfrak{g}_{\mathbb{Q}}$
defined over $\mathbb{Q}$ such that
$\mathfrak{g}\cong\mathfrak{g}_{\mathbb{Q}}\otimes{\mathbb{R}}$. A subalgebra
$\mathfrak{h}$ of $\mathfrak{g}$ is said to be rational with respect to a
given rational structure $\mathfrak{g}_{\mathbb{Q}}$ if
$\mathfrak{h}_{\mathbb{Q}}:=\mathfrak{h}\mathfrak{g}_{\mathbb{Q}}$ is a
rational structure for $\mathfrak{h}$.

It follows from a result of Malcev [M] that $\Gamma\backslash G$ is compact,
where G is a simply connected k-step nilpotent Lie group admitting a basis of
left invariant 1-forms for which the coefficients in the structure equations
are rational numbers, and $\Gamma$ is a lattice in $G$ of maximal rank (i.e.,
a discrete uniform subgroup, cf. [Rag]). Such a lattice $\Gamma$ exists in $G$
if and only if the Lie algebra $\mathfrak{g}$ of $G$ has a rational structure.
Indeed, If $\Gamma$ is a lattice of $G$ then its associated rational structure
is given by the $\mathbb{Q}$-span of $\log\Gamma$.

An invariant complex structure $J$ on a nilmanifold $\Gamma\backslash G$ is
called rational if it is compatible with the rational structure of $G$ , i.e.
$J(\mathfrak{g}_{\mathbb{Q}})\subseteq\mathfrak{g}_{\mathbb{Q}}.$

Remark 4.3: Let $M=\Gamma\backslash G$ be a complex nilmanifold of complex
dimension $n$ endowed with an invariant rational complex structure $J$.
Consider the surjective homomorphism
$\mathfrak{g}\rightarrow\mathfrak{g}/\mathfrak{g}^{1}_{J}$, where
$\mathfrak{g}^{1}_{J}=\mathfrak{g}^{1}+J\mathfrak{g}^{1}$. Let $G$,
$G^{1}_{J}$ and $K$ be the simply connected Lie groups respectively with Lie
algebra $\mathfrak{g}$, $\mathfrak{g}^{1}_{J}$ and
$\mathfrak{g}/\mathfrak{g}^{1}_{J}$, then we have the surjective homomorphism

$p:G\rightarrow K,$

with $K$ abelian. Since $J$ is rational then, by [CF] $\mathfrak{g}^{1}_{J}$
is a rational subalgebra of $\mathfrak{g}$. Then $\Gamma^{1}:=\Gamma\cap
G^{1}_{J}$ is a uniform discrete subgroup of $G^{1}_{J}$ [CG], Theorem 5.1.11.
By [CG, Lemma 5.1.4 (a)], $p(\Gamma)$ is a uniform discrete subgroup of $K$
(i.e. $p(\Gamma)\backslash K$) is compact, cf. [Rag]). By Lemma 2 in [CF] the
map

$\tilde{p}:\Gamma\backslash G\rightarrow p(\Gamma)\backslash K$

is a holomorphic fibre bundle. Moreover, since $K$ is abelian,
$p(\Gamma)\backslash K$ is a complex torus $\mathbb{T}$ of complex dimension
$\mathfrak{H}^{1}(M)=n-\dim_{{\mathbb{C}}}\mathfrak{g}^{1}_{J}$. Therefore, if
$\mathfrak{H}^{1}(M)=1$, the torus $\mathbb{T}$ is algebraic, and by 4 and 1.2
we have $a(M)=1.$

Remark 4.4:  Let $M=\Gamma\backslash G$ be a complex nilmanifold of complex
dimension $n$ endowed with an invariant complex structure $J$ such that
$J\mathfrak{g}^{1}=\mathfrak{g}^{1}$, then $M$ is the total space of a
holomorphic fibration $\pi:M\rightarrow\mathbb{T}$, with $\mathbb{T}$ a
complex torus of complex dimension
$\mathfrak{H}^{1}(M)=n-\dim_{{\mathbb{C}}}\mathfrak{g}^{1}$. Therefore, if
$\mathbb{T}$ is algebraic, i.e. $a(\mathbb{T})=\mathfrak{H}^{1}(M)$, we have
by 4 and 1.2 we have $\mathfrak{a}(B)=\mathfrak{a}(M)=\mathfrak{H}^{1}(M).$

Note that if $J$ is bi-invariant, i.e. if $M=\Gamma\backslash G$ is complex
parallelalizable then $J\mathfrak{g}^{1}=\mathfrak{g}^{1}$. For a general
result on the algebraic dimension of complex parallelalizable solvmanifolds
see Theorem 2 and its Corollary in [Sak].

We will apply the previous remarks to complex nilmanifolds of complex
dimension $3$.

Example 4.5: Let $J$ be a complex structure on a real $6$-dimensional
nilpotent Lie algebra. For the notion of “nilpotent complex structure” on a
nilmanifold, please see [Ug]. By [Ug] the complex structure $J$ is either
nilpotent or non-nilpotent and

  1. (a)

If $J$ is non nilpotent, then there is a basis of $(1,0)$-forms
$(\omega^{1},\omega^{2},\omega^{3})$ such that

$\left\\{\begin{array}[]{l}d\omega^{1}=0,\\\\[3.0pt]
d\omega^{2}=E\,\omega^{1}\wedge\omega^{3}+\omega^{1}\wedge\overline{\omega}^{3},\\\\[3.0pt]
d\omega^{3}=A\,\omega^{1}\wedge\overline{\omega}^{1}+ib\,\omega^{1}\wedge\overline{\omega}^{2}-ib\overline{E}\,\omega^{2}\wedge\overline{\omega}^{1},\end{array}\right.$

where $A,E\in{\mathbb{C}}$ with $|E|=1$ and $b\in{\mathbb{R}}-\\{0\\}$.

  2. (b)

If $J$ is nilpotent, then there is a basis of $(1,0)$-forms
$(\omega^{1},\omega^{2},\omega^{3})$ satisfying

$\left\\{\begin{array}[]{lcl}d\omega^{1}&=&0,\\\\[3.0pt]
d\omega^{2}&=&\varepsilon\omega^{1}\wedge\overline{\omega}^{1},\\\\[3.0pt]
d\omega^{3}&=&\rho\,\omega^{1}\wedge\omega^{2}+(1-\varepsilon)A\,\omega^{1}\wedge\overline{\omega}^{1}+B\,\omega^{1}\wedge\overline{\omega}^{2}\\\\[2.0pt]
&&+C\,\omega^{2}\wedge\overline{\omega}^{1}+(1-\varepsilon)D\,\omega^{2}\wedge\overline{\omega}^{2},\end{array}\right.$

where $A,B,C,D\in{\mathbb{C}}$ and $\varepsilon,\rho\in\\{0,1\\}$.

Suppose that the real and imaginary parts of the complex structure equations
constants are rational, then $G$ admits a lattice $\Gamma$. Let
$M=\Gamma\backslash G$ be the compact quotient endowed with the induced
invariant complex structure $J$.

In the case $(a)$ we have that $\mathfrak{H}^{1}(M)=1$. In the case $(b)$ we
have the following cases:

  1. (b1)

$\mathfrak{H}^{1}(M)=1$ if $\varepsilon=1$ and $\rho^{2}+|B|^{2}+|C|^{2}\neq
0$

  2. (b2)

$\mathfrak{H}^{1}(M)=2$ if $\varepsilon=0$

  3. (b3)

$\mathfrak{H}^{1}(M)=2$ if if $\varepsilon=1$ and $\rho=B=C=0$

Therefore, in the cases $(a)$ and $(b1)$, since $J$ is rational, by 1.2 and
previous remarks, we have $a(M)=\mathfrak{H}^{1}(M)=1$. In the case $(b3)$,
$G$ is the direct product of the real $3$-dimensional Heisenberg group by
$\mathbb{R}^{3}$.

In the case $(b2)$, $\rho^{2}+|B|^{2}+|C|^{2}+|D^{2}|=0$, then $G$ is the
direct product of the real $3$-dimensional Heisenberg group by
$\mathbb{R}^{3}$. If $\rho^{2}+|B|^{2}+|C|^{2}+|D^{2}|\neq 0$, then
$J\mathfrak{g}^{1}=\mathfrak{g}^{1}$ is a rational subalgebra of complex
dimension $1$ and $M$ is the total space of a holomorphic fibre bundle over a
complex torus $\mathbb{T}$ of complex dimension $2$. Therefore, if
$\mathbb{T}$ is algebraic then by previous remarks we have
$a(M)=\mathfrak{H}^{1}(M)=2$.

An explicit example of the case (b2) is given by the well known Iwasawa
manifold $M$. The Iwasawa manifold $M$ is defined as the quotient
$\Gamma\backslash G$, where

$G=\left\\{\left(\begin{array}[]{ccc}1&z_{1}&z_{3}\\\ 0&1&z_{2}\\\
0&0&1\end{array}\right)\,\mid\,z_{i}\in\mathbb{C}\right\\}$

is the complex Heisenberg group and $\Gamma$ is the lattice defined by taking
$z_{i}$ to be Gaussian integers, acting by left multiplication. The 1-forms

$\omega^{1}=dz_{1},\omega^{2}=dz_{2},\omega^{3}=-dz_{3}+z_{1}dz_{2}$

are left-invariant on $G$. Define a r basis $(e^{1},...,e^{6})$ of real
1-forms by setting

$\omega^{1}=e^{i}+ie^{2},\omega^{2}=e^{3}+ie^{4},\omega^{6}=e^{5}+ie^{6}.$

These 1-forms are pullbacks of corresponding 1-forms on the compact quotient
$M$, which we denote by the same symbols and they satisfy the structure
equations

$\begin{array}[]{l}de^{j}=0,\,j=1,2,3,4,\\\ de^{5}=e^{1}\wedge
e^{3}-e^{2}\wedge e^{4},\\\ de^{6}=e^{1}\wedge e^{4}+e^{2}\wedge
e^{3}.\end{array}$

The Iwasawa manifold $M$ is the total space of a principal $T^{2}$-bundle over
the real torus $T^{4}$. The mapping $p:M\rightarrow T^{4}$ is induced from the
projection $(z_{1},z_{2},z_{3})\mapsto(z_{1},z_{2})$ and the space of
invariant 1-forms annihilating the fibres of p is given by $\operatorname{\sf
span}\langle e^{1},e^{2},e^{3},e^{4}\rangle$

Then $p:M\rightarrow T^{4}$ induces a complex structure $\hat{J}$ on the real
4-dimensional torus $T^{4}$ such that $p:(M,J)\rightarrow(T^{4},\hat{J})$ is
holomorphic.

Claim 4.6:  For the invariant complex structures on the Iwasawa manifold, $M$,
$a(M)=a(T^{4},\hat{J})$

Proof: From 1.1, any meromorphic function is constant on the fibers of the
projection $M{\>\longrightarrow\>}(T^{4},\hat{J})$. This implies that
$a(M)=a(T^{4},\hat{J})$.

Now consider the possible algebraic dimension of $M$. First note that not all
complex structures on the base $T^{4}$ arise as $\hat{J}$. A description of
the set of such $J$ is given in [KS] and it is known that it has 4 components.
We’ll use one of them to see that that invariant structures on the Iwasawa
nilmanifold can have algebraic dimension 0, 1 and 2. If we start with the
canonical structure on $T^{4}$ which corresponds to the standard lattice in
$R^{4}$ it gives the case of algebraic dimension 2. From [KS] formula (10) we
know that the matrix $X=\left(\begin{array}[]{ll}0&\sqrt{2}-i\sqrt{3}\\\
0&0\\\ \end{array}\right)$ corresponds to a complex structure on $T^{4}$ which
is $\hat{J}$ for some $J$ on $M$. On the other side, the period matrix $\tau$,
as explained above is $iId+2X$ and has $\tau=x+iy$ with
$x=\left(\begin{array}[]{ll}0&2\sqrt{2}\\\ 0&0\\\ \end{array}\right)$ and
$y=\left(\begin{array}[]{ll}1&-2\sqrt{3}\\\ 0&1\\\ \end{array}\right)$. So for
such $\tau$ the algebraic dimension of the base is 1. In particular we have
strict inequalities in 1.2. Again, the base generically has algebraic
dimension 0, which leads to vanishing of the algebraic dimension of $J$ on the
Iwasawa nilmanifold.

Example (Compact Lie groups) It is well known that every even-dimensio nal
compact Lie group $G$ admits an invariant complex structure ([Sam]). The
construction uses the structure theory for semisimple Lie algebras and
provides a holomorphic fibration $G\rightarrow Fl$ to the complete flag
manifold $Fl=G/T$, where $T$ is a maximal torus in $G$. The manifold $Fl$ is
algebraic and its algebraic dimension is equal to its complex dimension. On
the other side, $\mathfrak{g}^{ss}=\mathfrak{g}^{\prime}$, so the space of
holomorphic differentials is trivial. Hence by 4, $a(M)\geqslant
dim(Fl)>\mathfrak{H}^{1}(M)=0$. Similarly, for non-Kähler compact complex
homogeneous spaces $G/H$ with $G$ compact, the inequality
$a(M)\leqslant\mathfrak{H}^{1}(M)$ does not hold in general.

Acknowledgements: The work on this project started when the first two authors
visited the HSE in Moscow. They are grateful for the hospitality and
stimulating environment at the ”Bogomolov Lab” there. Part of the work was
done while the second named author visited University di Torino (Torino), Max
Plank Institute for Mathematics (Bonn) and Institute of Mathematics at the
Bulgarian Academy of Sciences. He thanks all institutions for the hospitality.
The visits were partially funded by grants from G.N.S.A.G.A. and MPIM. The
third named author is thankful to Alexandra Victorova and Alexandra
Skripchenko for their ideas and many stimulating discussions of the subject.
We also thank S. Rollenske for his useful comments on the paper.

## References

  * [BD] M.L. Barberis, I. Dotti, Abelian complex structures on solvable Lie algebras, J. Lie Theory 14(1) (2004), 25–34.
  * [BDV] M. L. Barberis, I. G. Dotti, M. Verbitsky, Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry, Math. Res. Lett. 16 (2009), 331–347.
  * [B] F. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), 1–40.
  * [BG] C. Benson, C.S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27(4) (1988) 513–518.
  * [BL] C. Birkenhacke, H. Lange, Complex Tori, Progress in Mathematics 177, Birkhauser, 1999.
  * [Bouck] S. Boucksom. On the volume of a line bundle Intern. J. Math. 13, no 10 (2002) 1043 – 1063.
  * [Ca81] F. Campana, Coréduction algébrique d’un espace analytique faiblement kählérien compact, Invent. Math. 63 (1981) 187-223.
  * [CF] S. Console, A. Fino, Dolbeault cohomology of compact nilmanifolds, Transformation Groups 6 (2) (2001), 111–124.
  * [CFP] S. Console, A. Fino, Y.S. Poon, Stability of abelian complex structures, Internat. J. Math. 17 (2006), no. 4, 401–416.
  * [CFG1] L.A. Cordero, M. Fernández, A. Gray, Symplectic manifolds with no Kähler structure, Topology 25 (1986), no. 3, 375–380.
  * [CFG2] L.A. Cordero, M. Fernández, A. Gray, The Frölicher spectral sequence and complex compact nilmanifolds, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 17, 753–756.
  * [CFL] L.A. Cordero, M. Fernández, M. de León, Examples of compact complex manifolds with no Kähler structure, Portugal. Math. 44 (1987), no. 1, 49–62.
  * [CFGU] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte, Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology, Trans. Amer. Math. Soc. 352 (12), 5405–5433.
  * [CG] L.J. Corwin, F.P. Greenleaf, Representations of nilpotent Lie groups and their applications, Cambridge studies in advanced mathematics 18, Cambridge, New York, 1990.
  * [DP] J.-P. Demailly, M. Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Annals of Math. 159 (2004) 1247-1274.
  * [DF1] I. Dotti, A. Fino, Abelian hypercomplex 8-dimensional nilmanifolds, Ann. Global Anal. Geom. 18 (1) (2000), 47–59.
  * [FG] M. Fernández, A. Gray, The Iwasawa manifold, Differential geometry, Peñíscola 1985, 157-159, Lecture Notes in Math., 1209, Springer, Berlin, 1986.
  * [FG] A. Fino, G. Grantcharov, On some properties of the manifolds with skew-symmetric torsion and holonomy $SU(n)$ and $Sp(n)$, Adv. Math. 189 (2004), no. 2, 439–450.
  * [G] H. Guggenheimer, Sur les variétés qui possèdent une forme extérieure quadratique fermée, C. R. Acad. Sci. Paris 232, (1951). 470-472.
  * [H] K. Hasegawa, Minimal models of nilmanifolds,  Proc. Amer. Math. Soc. 106 (1989), no. 1, 65–71.
  * [Jac] H. Jacobowitz, Transversely holomorphic foliations and CR structures. VI Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 1999). Mat. Contemp. 18 (2000), 175-194.
  * [KS] G. Ketsetzis, S. Salamon, Complex structures on the Iwasawa manifold, Adv. Geom. 4 (2004), no. 2, 165-179.
  * [M] A. I. Mal’čev, On a class of homogeneous spaces, AMS Translation No. 39 (1951).
  * [Mi] J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976), 293–329.
  * [N] K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. 59 (1954), 531–538.
  * [Pet] Th. Peternell, Modifications, Several complex variables, VII, 285-317, Encyclopaedia Math. Sci., 74, Springer, Berlin, (1994)
  * [Rag] M. S. Raghunathan, Discrete subgroups of Lie groups, Springer E.M.G. 68, Berlin, Heidelberg, New York, 1972.
  * [R] S. Rollenske, Geometry of nilmanifolds with left-invariant complex structure and deformations in the large, Proc. Lond. Math. Soc. (3) 99 (2009), no. 2, 425-460.
  * [Sak] Y. Sakane, On compact complex parallelalisable solvmanifolds, Osaka J. Math. 13 (1976), 187-212.
  * [S] S.M. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311–333.
  * [Sam] H. Samelson, A class of compact-analytic manifolds, Portugaliae Math. 12 (1953), 129-132.
  * [T] W. P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), no. 2, 467-468.
  * [Ue75] K. Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics 439, Springer-Verlag, Berlin-New York (1975).
  * [Ug] L. Ugarte, Hermitian structures on six-dimensional nilmanifolds, Transform. Groups 12 (2007), 175–202.
  * [W] J. Winkelmann, Complex Analytic Geometry of Complex Parallelizable Manifolds, Mem. Soc. Math. France (N.S.) 72-73, Soc. Math. France, Montrouge, 1998.

Anna Fino
Dipartimento di Matematica G. Peano
Universitá di Torino
via Carlo Alberto 10, 10123 Torino, Italy
<EMAIL_ADDRESS>

Gueo Grantcharov
Department of Mathematics and Statistics
Florida International University
Miami Florida, 33199, USA
<EMAIL_ADDRESS>

Misha Verbitsky
Laboratory of Algebraic Geometry,
National Research University Higher School of Economics,
Department of Mathematics, 7 Vavilova Str. Moscow, Russia,
<EMAIL_ADDRESS>also:
Université Libre de Bruxelles, CP 218,
Bd du Triomphe, 1050 Brussels, Belgium