1. Introduction
Nilmanifolds constitute a well-known class of compact manifolds providing interesting explicit examples of geometric structures with special properties. A nilmanifold is a compact quotient
of a connected and simply connected nilpotent Lie group
G by a lattice
of maximal rank in
G. Hence, any left-invariant geometric structure on
G descends to
N. We will refer to such structures as invariant. For instance, there are nilmanifolds admitting invariant complex structures, as the Iwasawa manifold, or invariant symplectic forms, as the Kodaira–Thurston manifold, with remarkable properties [
1,
2]. However, by [
3], a nilmanifold cannot admit any Kähler metric (invariant or not), unless it is a torus. Since there are also nilmanifolds with no invariant complex structures or symplectic forms, it is an interesting problem to understand which nilmanifolds do admit such kinds of structures.
Symplectic and complex geometries constitute two special cases in the unified framework given by generalized complex geometry, introduced by Hitchin in [
4] and further developed by Gualtieri [
5]. In [
6], Cavalcanti and Gualtieri study invariant generalized complex structures on nilmanifolds. Furthermore, Angella, Calamai and Kasuya show in [
7] that nilmanifolds provide a nice class for investigating cohomological aspects of generalized complex structures.
In ([
6], Theorem 3.1) the authors prove that any invariant generalized complex structure on a
-dimensional nilmanifold must be generalized Calabi–Yau, extending a result of Salamon [
8] for invariant complex structures. This means that any generalized complex structure is given by a (left-invariant) trivialization
of the canonical bundle, i.e.,
where
are real invariant 2-forms and
is a globally decomposable complex
k-form, i.e.,
, where each
is an invariant 1-form. Moreover, these data satisfy the non-degeneracy condition
as well as the integrability condition
The integer
is called the type of the generalized complex structure. Type
corresponds to usual complex structures, whereas structures of type
are symplectic. It is proved in [
6] that every 6-dimensional nilmanifold admits a generalized complex structure of type
k, for at least one
; however, it is shown that there are nilmanifolds in eight dimensions not admitting any invariant generalized complex structure.
Let
be the nilpotent Lie algebra underlying the nilmanifold
, and let
be the Chevalley–Eilenberg complex seen as a commutative differential graded algebra (CDGA). Hasegawa proved in [
3] that this CDGA provides not only the
-minimal model of
N but also its
-minimal model. A result by Bazzoni and Muñoz asserts that, in six dimensions, there are infinitely many rational homotopy types of nilmanifolds, but only 34 different real homotopy types (see [
9], Theorem 2). Hence, there only exits a finite number of real homotopy types of 6-dimensional nilmanifolds admitting any kind of geometric structure. The existence of infinitely many real homotopy types of 8-dimensional nilmanifolds with a complex structure (having special Hermitian metrics) is proved in [
10]. However, these nilmanifolds do not admit any symplectic form. For this reason, we address the problem of finding a family of nilmanifolds with infinitely many real homotopy types that admit not only complex and symplectic structures, but also generalized complex structures of every possible type. In particular, we prove the following:
Theorem 1. There are infinitely many real homotopy types of 8-dimensional nilmanifolds admitting generalized complex structures of every type k, for .
Although in dimension 6 there are nilmanifolds admitting generalized complex structures of every possible type, their real homotopy types are finite. As our result shows, this does no longer hold in higher dimensions.
This paper is structured as follows. In
Section 2, we review some general results about minimal models and homotopy theory, and we define a family of nilmanifolds
in eight dimensions depending on a rational parameter
.
Section 3 is devoted to the construction of generalized complex structures on the nilmanifolds
. More concretely, we prove the following:
Proposition 1. For each , the 8-dimensional nilmanifold has generalized complex structures of type k, for every .
In
Section 4, we study the real homotopy types of the nilmanifolds in the family
. More precisely, the result below is attained:
Proposition 2. If , then the nilmanifolds and have non-isomorphic -minimal models, so they have different real homotopy types.
Note that Theorem 1 is a direct consequence of Propositions 1 and 2. Moreover, by taking products with even dimensional tori, the result holds in any dimension
and for every
. Furthermore, our result in Theorem 1 can be extended to the complex homotopy setting (see Remark 2 for details). Since the nilmanifolds
cannot admit any Kähler metric [
3], one has the following:
Corollary 1. Let . There are infinitely many complex homotopy types of -dimensional compact non-Kähler manifolds admitting generalized complex structures of every type k, for .
2. The Family of Nilmanifolds
Let us start recalling some general results about homotopy theory and minimal models, with special attention to the class of nilmanifolds. In [
11], Sullivan shows that it is possible to associate a minimal model to any nilpotent CW-complex
X, i.e., a space
X whose fundamental group
is a nilpotent group that acts in a nilpotent way on the higher homotopy group
of
X for every
. Recall that a minimal model is a commutative differential graded algebra, CDGA for short,
defined over the rational numbers
and satisfying a certain minimality condition, that encodes the rational homotopy type of
X [
12].
More generally, let be the field or . A CDGA defined over is said to be minimal if the following conditions hold:
- (i)
is the free commutative algebra generated by the graded vector space ;
- (ii)
there exists a basis , for some well-ordered index set J, such that for , and each is expressed in terms of the preceding .
A
-minimal model of a differentiable manifold
M is a minimal CDGA
over
together with a quasi-isomorphism
from
to the
-de Rham complex of
M, i.e., a morphism
inducing an isomorphism in cohomology. Here, the
-de Rham complex of
M is the usual de Rham complex of differential forms
when
, whereas for
one considers
-polynomial forms instead. Notice that the
-minimal model is unique up to isomorphism, since
. By [
11,
13], two nilpotent manifolds
and
have the same
-homotopy type if and only if their
-minimal models are isomorphic. It is clear that if
and
have different real homotopy types, then
and
also have different rational homotopy types.
Let N be a nilmanifold, i.e., is a compact quotient of a connected and simply connected nilpotent Lie group G by a lattice of maximal rank. For any nilmanifold N, one has , which is nilpotent, and for every . Therefore, nilmanifolds are nilpotent spaces.
Let
n be the dimension of the nilmanifold
, and let
be the Lie algebra of
G. It is well known that the minimal model of
N is given by the Chevalley–Eilenberg complex
of
. Recall that by [
14], the existence of a lattice
of maximal rank in
G is equivalent to the nilpotent Lie algebra
being rational, i.e., there exists a basis
for the dual
such that the structure constants are rational numbers. Thus, the rational and the nilpotency conditions of the Lie algebra
allow to take a basis
for
satisfying
with structure constants
.
Therefore,
is a CDGA satisfying both conditions
(i) and
(ii) with ordered index set
and
, where
for
. That is to say, the CDGA
over
is minimal, and it is determined by
with
n generators
of degree 1 satisfying equations of the form (
4). Notice that the CDGA
over
is also minimal, since it is given by
There is a canonical morphism
from the Chevalley–Eilenberg complex
to the de Rham complex
of the nilmanifold. Nomizu proves in [
15] that
induces an isomorphism in cohomology, so the
-minimal model of the nilmanifold
is given by (
6). Hasegawa observes in [
3] that (
5) is the
-minimal model of
N and that, conversely, given a
-minimal CDGA of the form (
5), there exists a nilmanifold
N with (
5) as its
-minimal model.
Deligne, Griffiths, Morgan and Sullivan prove in [
13] that the
-minimal model,
or
, of a compact Kähler manifold is formal, i.e., it is quasi-isomorphic to its cohomology. Hasegawa shows in [
3] that the minimal model (
5) is formal if and only if all the structure constants
in (
4) vanish, so a symplectic nilmanifold does not admit any Kähler metric unless it is a torus. (See for instance [
2,
16] for more results on homotopy theory and applications to symplectic geometry.)
Bazzoni and Muñoz study in [
9] the
-homotopy types of nilmanifolds of low dimension. They prove that, up to dimension 5, the number of rational homotopy types of nilmanifolds is finite. However, in six dimensions the following result holds:
Theorem 2. ([9], Theorem 2) There are infinitely many rational homotopy types of 6-dimensional nilmanifolds, but there are only 34 real homotopy types of 6-dimensional nilmanifolds. As a direct consequence, there is only a finite number of real homotopy types of 6-dimensional nilmanifolds admitting an extra geometric structure of any kind (in particular, generalized complex structures; see
Section 3 for definition). We will prove that, in contrast to the 6-dimensional case, there are infinitely many real homotopy types of 8-dimensional nilmanifolds admitting generalized complex structures of every type
k, for
.
To construct such nilmanifolds, let us take a positive rational number
and consider the connected, simply connected, nilpotent Lie group
corresponding to the nilpotent Lie algebra
defined by
where
, being
a basis for
, and
. It is clear from (
7) that the Lie algebra
is rational, hence by the Mal’cev theorem [
14], there exists a lattice
of maximal rank in
. We denote by
the corresponding compact quotient.
Therefore, we have defined a family of nilmanifolds
of dimension 8 depending on the rational parameter
. We will study the properties of
, for
, in
Section 3 and
Section 4. Here, we simply provide their Betti numbers.
A direct calculation using Nomizu’s theorem [
15] allows to explicitly compute the de Rham cohomology groups of any nilmanifold
. In particular, for degrees
, the
l-th de Rham cohomology groups
are
Let
denote the
l-th Betti number of
. By duality we have:
One can finally compute the Betti number
taking into account that the Euler–Poincaré characteristic
of a nilmanifold always vanishes, namely,
which implies
. In particular, we observe that the Betti numbers of the nilmanifolds
do not depend on
.
3. Generalized Complex Structures on the Nilmanifolds
Generalized complex geometry, in the sense of Hitchin and Gualtieri [
4,
5], establishes a unitary framework for symplectic and complex geometries. Let
M be a compact differentiable manifold of dimension
. Denote by
the tangent bundle and by
the cotangent bundle, and consider the vector bundle
endowed with the natural symmetric pairing
Recall that the Courant bracket on the space
is given by
where
and
respectively denote the Lie derivative and the interior product. A generalized complex structure on
M is an endomorphism
satisfying
whose
i-eigenbundle
is involutive with respect to the Courant bracket.
There is an action of
on
given by
Now, for a generalized complex structure
with
i-eigenbundle
L, one can define the canonical line bundle
as
Any
is a non-degenerate pure form, i.e., it can be written as
where
are real 2-forms and
is a complex decomposable
k-form, such that
The number
k is called the type of the generalized complex structure. Moreover, any
is integrable, i.e., there exists
satisfying
Notice that the converse also holds: if is a line bundle such that any is a non-degenerate pure form and any is integrable, then we have a generalized complex structure whose i-eigenbundle is .
In the case that K is a trivial bundle admitting a nowhere vanishing closed section, the generalized complex structure is called generalized Calabi–Yau.
Recall that if
J is a complex structure on
M then
is a generalized complex structure of type
n, and if
is a symplectic form on
M then
is a generalized complex structure of type 0. Near a regular point (i.e., a point where the type is locally constant), a generalized complex structure is equivalent to a product of a complex and a symplectic structure ([
5], Theorem 3.6).
In the case of a nilmanifold
N, Cavalcanti and Gualtieri proved in ([
6], Theorem 3.1) that any invariant generalized complex structure on
N must be generalized Calabi–Yau. Hence, it is given by a (left-invariant) trivialization
of the canonical bundle of the form (
1) satisfying the non-degeneracy condition (
2) and the integrability condition (
3).
Let us now prove Proposition 1, that is, each nilmanifold
has generalized complex structures of every type
k, for
. These structures will be explicitly described in terms of the global basis of invariant 1-forms
on
given in (
7). We begin providing a structure of type 4.
Generalized complex structure of type 4 (complex structure). We define the following complex 1-forms:
From Equation (
7), we get
Declaring the forms
to be of bidegree
, we obtain an almost complex structure
J on the nilmanifold
for every
. It follows from (
9) that
has no
component, so
J is integrable. Hence,
is a generalized complex structure of type 4.
Remark 1. The complex nilmanifold
has a holomorphic Poisson structure given by the holomorphic bivector
of rank two, where
is the dual basis of
(see [
6], (Theorem 5.1) for the existence of such a bivector on nilmanifolds). It is worth observing that
does not admit any (invariant or not) holomorphic symplectic structure: since the center of the Lie algebra
has dimension 1, the complex structure defined by (
8) is strongly non-nilpotent (see [
17] for properties on this kind of complex structure); by [
18], a strongly non-nilpotent complex structure on an 8-dimensional nilmanifold cannot support any holomorphic symplectic form.
Generalized complex structure of type 3. Let us consider
, with
,
and
, where
and
are the complex 1-forms given in (
8). It is clear that
so the non-degeneracy condition (
2) is satisfied. A direct calculation using (
9) shows
and
which implies that
, i.e., the integrability condition (
3) holds.
Therefore, the nilmanifolds have generalized complex structures of type 3.
Generalized complex structure of type 2. Recall that the action of a bivector
is given by
. If
J is a complex structure and
is a holomorphic Poisson structure of rank
l, then one can deform
J into a generalized complex structure of type
(see [
5]). In ([
6], Theorem 5.1) it is proved that every invariant complex structure on a
-dimensional nilmanifold can be deformed, via such a
-field with
, to get an invariant generalized complex structure of type
. Therefore, our nilmanifolds have a generalized complex structure of type 2.
More concretely, in view of Remark 1, from the generalized complex structure of type 4 defined by
above, we get that
with
, is a generalized complex structure of type 2. Indeed,
and
. Thus,
and
and
by the Equation (
9).
For the definition of generalized complex structures of type 1 and type 0 we will deal with the space
of invariant closed 2-forms on the nilmanifold
. The following lemma is straightforward:
Lemma 1. Every invariant closed 2-form ω on the nilmanifold is given bywhere . Hence, the space has dimension 12. Generalized complex structure of type 1. We must find
, with
real invariant 2-forms and
a complex 1-form, satisfying
Since
is a complex 1-form, it can be written as
, for some complex coefficients
. Let us choose
, which satisfies
according to the structure Equation (
7).
We consider
and let
be any 2-form given in (
10) with coefficients
. A direct calculation shows that
Since both B and are closed, the condition is trivially satisfied, and defines a generalized complex structure of type 1 on the nilmanifold .
Generalized complex structure of type 0 (symplectic structure). The form
in Lemma 1 determined by (
10) satisfies
It suffices to choose, for instance, and to get a symplectic form on .
This concludes the proof of Proposition 1.
4. The Nilmanifolds and Their Minimal Model
The goal of this section is to prove Proposition 2, i.e., the nilmanifolds and have non-isomorphic -minimal models for .
As we recalled in
Section 2, the
-minimal model of the nilmanifold
is given by the Chevalley–Eilenberg complex
of its underlying Lie algebra
. Consequently, to prove Proposition 2, it suffices to show that the real Lie algebras
,
, define a family of pairwise non-isomorphic nilpotent Lie algebras. Indeed, we will prove the following:
Proposition 3. If the nilpotent Lie algebras and are isomorphic, then .
Remember that in eight dimensions, no classification of nilpotent Lie algebras is available. Indeed, nilpotent Lie algebras are classified only up to real dimension 7. More concretely, Gong classified in [
19] the 7-dimensional nilpotent Lie algebras in 140 algebras together with 9 one-parameter families. One can check that our family
is not an extension of any of those 9 families, i.e., the quotient of
by its center (which has dimension 1) does not belong to any of the 9 one-parameter families of Gong. Furthermore, the usual invariants for nilpotent Lie algebras are the same for all the algebras in the family
. For instance, the dimensions of the terms in the ascending central series are
, whereas those of the descending central series are
(see Lemma 3 for further details). Moreover, the dimensions of the Lie algebra cohomology groups
coincide for every
, as shown at the end of
Section 2. For this reason, we will directly analyze the existence of an isomorphism between any two of the nilpotent Lie algebras in our family
.
The following technical lemma will be useful for our purpose.
Lemma 2. Let be an isomorphism of the Lie algebras and . Consider an ideal , and let be the corresponding ideal in . Let , resp. , be a basis of , resp. , and complete it up to a basis of , resp. of . Denote the dual bases of and respectively by and . Then, the dual map satisfies Proof. Let
and
be the natural projections, and
the Lie algebra isomorphism induced by
f on the quotients. Taking the corresponding dual maps, we have the following commutative diagrams:
Taking the basis
of
, we have that
is a basis of
. Let
be its dual basis for
. Using a similar procedure, we find a basis
for
. Since the maps
and
are injective, and the diagram is commutative, we get
for any
. □
Applying the previous result to our particular case, we get:
Lemma 3. Consider and for . If is an isomorphism of Lie algebras, then in terms of their respective bases and given in (7), the dual map satisfies Proof. Recall that the ascending central series of a Lie algebra
is defined by
, where
and
Observe that is the center of .
Let
and
be the bases for
and
dual to
and
, respectively. In terms of these bases, the ascending central series of
and
are
and
Since
for any Lie algebra isomorphism
, applying Lemma 2 to the ideals
for
one gets (
12) for
and 7.
Moreover, the derived algebras of
and
are, respectively,
Using again Lemma 2 with
, we obtain (
12) for
. □
We are now in the conditions to prove Proposition 3.
Proof of Proposition 3. Given any homomorphism of Lie algebras
, its dual map
naturally extends to a map
that commutes with the differentials, i.e.,
. Hence, in terms of the bases
for
and
for
satisfying the Equations (
7) with respective parameters
and
, any Lie algebra isomorphism is defined by
satisfying conditions
where the matrix
belongs to
.
We first note that the preceding lemma allows us to simplify the matrix . In fact, from Lemma 3 one has that for and , for and , and also . Since belongs to , the previous conditions imply that and .
Note also that (
14) is trivially fulfilled for
. Hence, it suffices to focus on
. We will denote by
the coefficient for
in the expression of the 2-form
.
By a direct calculation we have
Since
, we conclude that
. Now observe that the following expressions must annihilate:
Solving
and
from the first two equations and replacing their values in the last ones, we get:
Moreover, the following terms must vanish:
From the second one, we have
. Since
, in particular also
. Using the first expression above, we can then solve
In addition, observe that
leads to
We now check that the vanishing of the coefficient
leads to a contradiction. Indeed, in such a case, the first expression in (
17) becomes
, and from (
15) we then have
, which plugged into (
16) gives
. Replacing this value in the second equation of (
17), the condition
arises. Since
and
are greater than zero, we are forced to consider
. However, this leads to
, which is a contradiction. Hence, we necessarily have that
is nonzero.
Since
, the condition
implies
. Replacing this value in (
15), (
16), and (
17), we obtain:
As , one immediately has and . Consequently , which allows us to conclude , and thus . This completes the proof of the proposition. □
Remark 2. In addition to the notions of rational and real homotopy types, there is also the notion of complex homotopy type [
13]. Two manifolds
X and
Y have the same
-homotopy type if and only if their
-minimal models
and
are isomorphic. Here,
and
are the rational minimal models of
X and
Y, respectively. Recall that when the field
has
, the
-minimal model is unique up to isomorphism. Clearly, if
X and
Y have different complex homotopy types, then
X and
Y have different real (hence, also rational) homotopy types.
For nilmanifolds, it is proved in ([
9], Theorem 2) that there are exactly 30 complex homotopy types of 6-dimensional nilmanifolds. It is worth remarking that if
, then our nilmanifolds
and
have different
-minimal models. Indeed, it can be checked that the proof of Proposition 2 above directly extends to the case when the matrix
defined in (
13) belongs to
. In conclusion, our main result in Theorem 1 extends to the complex case, i.e., there are infinitely many complex homotopy types of 8-dimensional nilmanifolds admitting a generalized complex structure of ever type
k, for
. Now, Corollary 1 is a consequence of the fact that the product nilmanifolds
do not admit any Kähler metric.