“garcia de galdeano”

garcía de galdeano

seminariomatemáticon. 4

PRE-PUBLICACIONES delseminario matematico 2002

L. A. CorderoM. FernandezL. Ugarte

Lefschetz complex conditionsfor complex manifolds

Universidad de Zaragoza

Lefschetz complex conditions for complex manifolds

Luis A. Cordero, Marisa Fernandez, Luis Ugarte

Abstract.- For any compact complex manifold M with a compatible symplectic form ω, we considerthe homomorphisms L1,0: H1,0(M) −→ Hn,n−1(M) and L0,1: H0,1(M) −→ Hn−1,n(M) given by thecup product with [ω]n−1, n being the complex dimension of M and H∗,∗(M) the Dolbeault cohomologyof M . We say that M has Lefschetz complex type (1, 0) (resp. (0, 1)) if L1,0 (resp. L0,1) is injective. Suchconditions can be considered as complex versions of the (real) Lefschetz condition studied by Benson-Gordon in [2] for symplectic manifolds. Within the class of compact complex nilmanifolds, we provethat the injectivity of L1,0 characterizes those complex structures which are abelian in the sense ofBarberis-Dotti-Miatello [1]. In contrast, complex tori are the only nilmanifolds having Lefschetz complextype (0, 1).

1 Introduction

Nowadays many results on the problem of constructing symplectic manifolds with no Kahlerianstructure (Thurston-Weinstein problem) are known. In particular, Benson-Gordon [2] andHasegawa [11] have completely solved this problem for the class of compact nilmanifolds bymeans of two different approaches.

Benson and Gordon proved that a symplectic nilmanifold satisfying the Lefschetz conditionis diffeomorphic to a torus, and since any compact Kahler manifold satisfies the hard Lefschetztheorem it follows that a non-toral symplectic nilmanifold cannot carry Kahler structures at all.

Hasegawa approach relies on the study of the formality of a minimal model for the de Rhamcohomology of a nilmanifold Γ\G. In fact, he proved that the minimal model given by Nomizutheorem [15] is never formal unless Γ\G be a torus. Thus, as a minimal model of a compactKahler manifold must be formal [7], a compact nilmanifold (not a torus) has no Kahler structure.

Recently, Lupton and Oprea [12] have given a rational homotopy theoretic proof of Benson-Gordon theorem. Since a Kahler manifold is, in particular, a complex manifold, it can beexpected that another approach to the solution of the Thurston-Weinstein problem might comeby exploiting the Dolbeault homotopy theory introduced in [14], which is in certain sense aversion of Sullivan theory for the Dolbeault cohomology of complex manifolds.

Our goal in this paper is the study of Lefschetz complex conditions for a large class of compactcomplex nilmanifolds Γ\G, namely the one constituted by those Γ\G for which their complexstructure is nilpotent in the sense of [6] (see Section 2 for details). Moreover, for such manifolds

02000 Mathematics Subject Classification. Primary 53C15, 53D05; Secondary 17B30, 53C30, 53C56.

Key words: complex manifold, Dolbeault cohomology, abelian complex structure, complex parallelizable nilman-

ifold, symplectic form.

1

we also present versions of both Benson-Gordon and Hasegawa theorems from the Dolbeaulthomotopy theoretic point of view (Theorems 3.3 and 4.1, respectively).

In order to formulate the Lefschetz complex conditions that we are interested in, let us remindfirst that a 2n-dimensional compact symplectic manifold (M,ω) is said to have Lefschetz typeif the homomorphism

L: H1(M) −→ H2n−1(M)(1)

given by the cup product L([α]) = [α][ω]n−1, for [α] ∈ H1(M), is injective. Actually L is anisomorphism because the de Rham cohomology groups H1(M) and H2n−1(M) are isomorphicby Poincare duality.

Consider now a triple (M, J, ω) given by a compact 2n-dimensional manifold M endowedwith a complex structure J that carries a compatible symplectic form ω, that is, ω has bidegree(1,1) with respect to J or equivalently ω(JX, JY ) = ω(X,Y ) for all vector fields X, Y on M .Then there exists a metric on M , not necessarily positive definite, given by g(X,Y ) = ω(JX, Y ).As we want to study the influence of the additional complex invariants arising in this case, ourcomplex analogs to the Lefschetz condition (1) will be described in terms of the Dolbeaultcohomology of M .

Let H∗,∗(M) denote the Dolbeault cohomology of (M, J). If (M, J, ω) is a Kahler manifold,i.e. if the corresponding metric g is positive definite, then the Hodge decomposition Hk(M) ∼=⊕p+q=kH

p,q(M) implies that the isomorphism (1) expresses as

L:H1,0(M)⊕H0,1(M) −→ Hn,n−1(M)⊕Hn−1,n(M)

and, since ω is of type (1,1) with respect to J , the map L splits into two isomorphisms:

L1,0: H1,0(M) −→ Hn,n−1(M) and L0,1: H0,1(M) −→ Hn−1,n(M).

In Theorem 3.3 we prove that complex tori are the only compact nilmanifolds for whichthe mapping L0,1 given by the cup product by [ω]n−1 is a monomorphism; this result can beconsidered as a Dolbeault homotopy theoretic version of Benson-Gordon theorem. In contrast,the injectivity of the homomorphism L1,0 above characterizes the nilpotent complex structureson nilmanifolds that are abelian in the sense of [1], as it is shown in Theorem 3.4.

The main tool in the proofs of these results is the Dolbeault homotopy theory developedin [6, 14], in particular the use of certain derivations on the Dolbeault minimal models ofnilmanifolds that we introduce in Section 2.

In Section 4 we prove a Dolbeault homotopy theoretic version of Hasegawa theorem, namelya compact nilmanifold with nilpotent complex structure is Dolbeault formal if and only if it isa complex torus, and we discuss some consequences of this fact.

Section 5 is devoted to analyze some properties of abelian complex structures on nilmanifolds.In Theorem 5.1 we show that, for any real dimension 2n, there exist compact nilmanifolds withabelian complex structure carring a compatible symplectic form. This existence result differsdramatically from Theorem 3.2, where it is proved that complex tori are the only compactcomplex parallelizable nilmanifolds with compatible symplectic structure, which extends thewell-known result of [9] for the Iwasawa manifold.

Finally, let us remark that it is not clear how to extend our results for the homomorphismsL1,0 and L0,1 to the general case of compact complex manifolds, nor even for compact complex

2

nilmanifolds whose complex structure is non-nilpotent. However, there are some reasons toconjecture that on a compact nilmanifold any complex structure, coming from a left invariantone, must be necessarily nilpotent in presence of a compatible symplectic form; so, if this is truethen Theorems 3.3 and 3.4 hold without the nilpotency restriction on the complex structure.

2 Derivations on Dolbeault minimal models of compact

nilmanifolds with nilpotent complex structures

Compact complex nilmanifolds have deserved special attention during the last years, becausethey provide examples of compact complex manifolds possessing interesting unusual properties(see the papers listed in the bibliography). A special class of such nilmanifolds is the oneconstituted by those compact nilmanifolds Γ\G endowed with a nilpotent complex structure(NCS for short), which is introduced in [6] and defined as follows.

Let G be a real nilpotent Lie group endowed with a left invariant complex structure J ,that is, J is an endomorphism of the Lie algebra g of G satisfying J2 = −Id and [JX, JY ] =[X, Y ]+J [JX, Y ]+J [X,JY ], for all X, Y ∈ g. Then, there exists an ascending series {al(J)}l≥0

of the Lie algebra g, associated to J in a natural way, defined inductively by

a0(J) = {0} , al(J) = {X ∈ g | [X, g] ⊆ al−1(J) and [JX, g] ⊆ al−1(J)} , l ≥ 1.

Notice that each J-invariant ideal al(J) of g satisfies al(J) ⊆ gl for l ≥ 0, gl being the termin the usual ascending central series {gl}l≥0 of g. A left invariant complex structure J on G iscalled nilpotent if the series {al(J)}l≥0 satisfies at(J) = g for some integer t > 0.

Two special and important classes of NCSs are the following. On one hand, if J makes G

a complex Lie group, i.e. [JX, Y ] = J [X, Y ] for all X, Y ∈ g, then al(J) = gl for all l ≥ 0; inthis case we shall say that J is complex parallelizable. On the other hand, J is called abelian if[JX, JY ] = [X, Y ] for all X,Y ∈ g, which implies again that J preserves the ascending centralseries of g (abelian hypercomplex structures are introduced and studied in [1, 8]).

Let us remark that although many nilpotent Lie groups have NCSs, however there exist Liegroups with no NCS but having left invariant complex structures. A complete classification upto dimension six is given in [18].

A compact nilmanifold with a NCS is by definition a complex manifold of the form Γ\G,a quotient of a simply-connected nilpotent Lie group G by a lattice Γ ⊂ G of maximal rank,whose complex structure is inherited from a NCS J on G by passing to the quotient. Weshall say that Γ\G is abelian or complex parallelizable if J is abelian or complex parallelizable,respectively. Notice that when G is a complex Lie group the compact nilmanifold Γ\G is complexparallelizable in the sense that there exist n holomorphic vector fields on Γ\G that are linearlyindependent at each point [22], where n is the complex dimension.

Let us recall next some basic definitions of the Dolbeault homotopy theory of Neisendorferand Taylor [14].

A differential bigraded algebra, DBA for short, M = (M∗,∗, ∂) is a bigraded commutativealgebra M∗,∗ over C with a differential ∂ of bidegree (0,1) which is a derivation with respectto the total degree. The Dolbeault complex (Λ∗,∗(M), ∂) of a complex manifold M provides an

3

example of DBA. The Dolbeault cohomology (H∗,∗(M), 0) of M with zero differential is againa DBA.

A DBAM = (M∗,∗, ∂) is a model for the Dolbeault complex of a complex manifold M if thereis a morphism ϕ : (M∗,∗, ∂) −→ (Λ∗,∗(M), ∂) of DBAs inducing an isomorphism on cohomology.If, in addition, the algebra M∗,∗ is free and there exists a collection of generators {xs}s∈S , forsome well ordered index set S, such that each ∂(xs) is expressed in terms of preceding generatorsxr (r < s) of total degree deg (xr) ≤ deg (xs), then M is said to be minimal.

Let G be a nilpotent Lie group with a left invariant complex structure J . Then the ex-terior bigraded algebra of complex valued left invariant differential forms on G, together withthe canonical differential ∂, is a DBA. In fact, this DBA is canonically identified to the DBA(Λ∗,∗(gC)∗, ∂), where gC denotes the complexification of the Lie algebra g of G, and the bigrad-uation on the exterior algebra Λ∗(gC)∗ is the one induced by J in a natural way. Thus, if Γ isa lattice in G of maximal rank then there is a canonical morphism of DBAs

i : (Λ∗,∗(gC)∗, ∂) −→ (Λ∗,∗(Γ\G), ∂).

Theorem 2.1 [6] Let Γ\G be a compact nilmanifold with a NCS. Then the morphism i in-duces an isomorphism on cohomology. Furthermore, (Λ∗,∗(gC)∗, ∂) is a minimal model for theDolbeault complex of Γ\G.

The minimality of the model is a consequence of the following characterization of a NCS interms of the structure equations of G. Let J be a left invariant complex structure on a nilpotentLie group G of real dimension 2n. In [6] it is proved that J is nilpotent if and only if there existsa basis {αi, αi; 1 ≤ i ≤ n} of left invariant complex 1–forms on G (where αi is of type (1,0)and hence its complex conjugate αi is of type (0,1) with respect to J) for which the structureequations have the form

dαi =∑

j<k<i

Aijk αj ∧ αk +∑

j,k<i

Bijk αj ∧ αk , 1 ≤ i ≤ n .(2)

Here the coefficients Aijk, Bijk are complex numbers.Let us now define integer numbers b1,0 and b0,1 which will play an important role in the

following. They are given as follows:

b1,0 = dim (g1,0/π1,0([g1,0, g0,1])) , b0,1 = dim(g0,1/[g0,1, g0,1]),

where π1,0 : gC = g1,0⊕g0,1 −→ g1,0 denotes the canonical projection. Since b1,0 = dimH1,0(gC)and b0,1 = dimH0,1(gC), from Theorem 2.1 we also have b1,0 = dim H1,0(Γ\G) = dimH1,0(M)and b0,1 = dimH0,1(Γ\G) = dimH0,1(M), where M is any model for the Dolbeault complex ofΓ\G. (Let us remark that Proposition 2.3 below implies that n ≥ b1,0 ≥ 1 and n ≥ b0,1 ≥ 2.)

Lemma 2.2

(i) A NCS J is complex parallelizable if and only if b1,0 = n.

(ii) A NCS J is abelian if and only if b0,1 = n.

In particular, g is an Abelian Lie algebra if and only if b1,0 = b0,1 = n.

4

Proof : It follows directly by taking into account that dα(X, Y ) = −α([X, Y ]), for all X,Y . Infact, J satisfies [JX, Y ] = J [X,Y ] for all X,Y ∈ g if and only if [g1,0, g0,1] = 0, or equivalentlyall the constants Bijk in (2) vanish. On the other hand, J is abelian, i.e. [JX, JY ] = [X, Y ] forall X, Y ∈ g if and only if [g0,1, g0,1] = 0, or equivalently all the constants Aijk = 0 in (2).

QED

Remark that from this proof it follows that J is abelian if and only if g0,1 and g1,0 are AbelianLie algebras.

From Theorem 2.1, denoting xi1,0 instead of αi and xi

0,1 instead of αi, we get:

Proposition 2.3 A minimal model for the Dolbeault complex of a compact nilmanifold Γ\Gwith a NCS J satisfying (2) is given by the DBA

M = (Λ∗,∗(x10,1, . . . , x

n0,1, x

11,0, . . . , x

n1,0), ∂),

where the generators have total degree 1 and bidegree as indexed, and ∂ is given by

∂xi0,1 =

∑

j<k<i

Aijk xj0,1x

k0,1 , ∂xi

1,0 =∑

j,k<i

Bijk xj1,0x

k0,1 , 1 ≤ i ≤ n.

Moreover, one can assume without loss of generality that the minimal model satisfies:

(i) ∂(x10,1) = · · · = ∂(xb0,1

0,1 ) = 0 and ∂(xk0,1) 6∈ 〈∂(x1

0,1), . . . , ∂(xk−10,1 )〉, for k = b0,1 + 1, . . . , n;

(ii) ∂(x11,0) = · · · = ∂(xb1,0

1,0 ) = 0 and ∂(xk1,0) 6∈ 〈∂(x1

1,0), . . . , ∂(xk−11,0 )〉, for k = b1,0 + 1, . . . , n.

Next we show some general results to be used in the following sections. The first result isstraightforward, so we omit the proof.

Lemma 2.4 Let M be a DBA as in Proposition 2.3. Then:

(i) ∂Mn,0 = 0, and thus dimHn,0(M) = 1;

(ii) ∂M0,n−1 = 0, and therefore dimH0,n(M) = 1.

Let M be the minimal DBA of Proposition 2.3. For each k, with 1 ≤ k ≤ n, we define aderivation θk

1,0:M∗,∗ −→M∗−1,∗, of bidegree (−1, 0), given on generators by

θk1,0(x

k1,0) = 1, θk

1,0(xi1,0) = θk

1,0(xj0,1) = 0,

for any i 6= k and j = 1, . . . , n.Similarly, a derivation θk

0,1 onM∗,∗ of bidegree (0,−1) is defined by the conditions θk0,1(x

k0,1) =

1 and θk0,1 = 0 for any other generator.

Lemma 2.5 The derivations θk1,0 and θk

0,1 satisfy:

(i) θk1,0∂ + ∂θk

1,0 + ρk = 0, where ρk is a derivation of bidegree (−1, 1) (and hence of totaldegree 0) which only depends on the constants Bikr, for k < i ≤ n and 1 ≤ r < i.

5

(ii) θk0,1∂ + ∂θk

0,1 + µk = 0, where µk is a bidegree (0, 0) derivation which only depends on theconstants Bijk and Airk, for 1 ≤ r < k < i ≤ n and 1 ≤ j < i, and on the constants Aiks,for k < s < i ≤ n.

In particular, ρn and µn are identically zero.

Proof : It is easy to check that ρk is given on generators by

ρk(xi1,0) = −

∑

r<i

Bikr xr0,1, k + 1 ≤ i ≤ n,

and ρk = 0 for any other generator. Similarly, the derivation µk is given on generators by

µk(xi1,0) =

∑

j<i

Bijk xj1,0, µk(xi

0,1) =∑

j<k<i

Aijk xj0,1 −

∑

k<r<i

Aikr xr0,1,

for k + 1 ≤ i ≤ n, and µk being zero for any other generator. QED

Corollary 2.6

(i) If ρk ≡ 0 for some k, then θk1,0 induces a derivation θk

1,0: H∗,∗(M) −→ H∗−1,∗(M) of

bidegree (−1, 0) on cohomology.

(ii) If µk ≡ 0 for some k, then θk0,1 induces a derivation θk

0,1:H∗,∗(M) −→ H∗,∗−1(M) of

bidegree (0,−1) on cohomology.

In particular, θn1,0 and θn

0,1 always induce derivations on cohomology.

The simultaneous vanishing of ρk and µk can be characterized as follows. Let {Zi, Zi; 1 ≤i ≤ n} be a basis of gC dual to the basis {αi, αi; 1 ≤ i ≤ n} of 1-forms satisfying (2). Takinginto account that dα(X,Y ) = −α([X, Y ]) for any α ∈ (gC)∗ and X, Y ∈ gC, and identifyingxk

1,0 ≡ αk and xk0,1 ≡ αk, we conclude that ρk and µk vanish simultaneously if and only if the

corresponding Zk ∈ (g1)C, where g1 is the center of the Lie algebra g. Notice that Zn alwaysbelongs to (g1)C.

Finally, the following result will be used in the next section.

Proposition 2.7 Let M be a DBA as in Proposition 2.3, and let ω ∈ M1,1 such that ωn 6= 0.Then the map ωr:M1,0 −→ Mr+1,r given by z 7→ zωr, for z ∈ M1,0, is injective for eachr = 1, . . . , n− 1.

Proof : Suppose that zωr = 0, where z =∑

akxk1,0. Since 0 = θk

1,0(zωr) = akωr − zθk

1,0(ωr),

we get akωr = rzωr−1θk

1,0(ω). Multiplying this equality by ωn−r and taking into account thatzωn−1 = zωrωn−1−r = 0, one concludes that akω

n = 0, i.e. ak = 0. Therefore, for each k thecoefficient ak = 0 and hence z = 0. QED

6

3 Lefschetz complex conditions

Let M be a compact complex manifold, of complex dimension n, endowed with a compatiblesymplectic structure ω, that is ω(JX, JY ) = ω(X, Y ) for all vector fields X, Y on M ; then ω isa real form of type (1, 1) which defines a nonzero class [ω] ∈ H1,1(M) satisfying [ω]n 6= 0.

Definition 3.1 We say that M has Lefschetz complex type (0, 1) (resp. (1, 0)) if the mappingL0,1:H0,1(M) −→ Hn−1,n(M) (resp. L1,0: H1,0(M) −→ Hn,n−1(M)) given by the cup productL0,1([α]) = [α][ω]n−1 (resp. L1,0([β]) = [β][ω]n−1) is a monomorphism.

Obviously, for any compact Kahler manifold both conditions hold. Notice that if a complexmanifold M has Lefschetz complex type (0, 1) then the Hodge numbers of M satisfy b0,1(M) ≤b1,0(M), whereas b1,0(M) ≤ b0,1(M) if L1,0 is a monomorphism.

Before going into the characterization of the compact nilmanifolds that satisfy Lefschetzcomplex condition of type (0, 1) we need to prove that complex tori are the only compact complexparallelizable nilmanifolds which admit indefinite Kahler metrics compatible with its naturalcomplex structure (compare with [9], where this fact was proved for the Iwasawa manifold).

Theorem 3.2 Let Γ\G be a compact complex parallelizable nilmanifold. If Γ\G is not a complextorus then Γ\G does not admit symplectic structure (homogeneous or otherwise) compatible withits complex parallelizable structure.

Proof : Since [g1,0, g0,1] = 0, Sakane theorem [17] implies that the second de Rham cohomologygroup decomposes as H2(Γ\G) ∼= E2,0

2 (Γ\G) ⊕ E1,12 (Γ\G) ⊕ E0,2

2 (Γ\G), where E1,12 (Γ\G) ∼=

H1(g1,0)⊗H1(g0,1) and E∗,∗r (Γ\G) are the terms in the Frolicher spectral sequence of Γ\G.

In this case we also have b0,1 = dim H1(g0,1) = dim(g0,1/[g0,1, g0,1]) = dim(g1,0/[g1,0, g1,0]) =dimH1(g1,0). Therefore, if {[α1

1,0], . . . , [αb0,1

1,0 ]} is a basis of H1(g1,0) then {[α10,1], . . . , [α

b0,1

0,1 ]} is abasis of H1(g0,1), αi

0,1 being the complex conjugate of αi1,0, 1 ≤ i ≤ b0,1. Now, any cohomology

class [ω] of type (1, 1) must be a linear combination of [αi1,0 ∧ αj

0,1], for 1 ≤ i, j ≤ b0,1. If Γ\G isnot a complex torus then b0,1 < n and [ω]n = 0, i.e. no closed form of bidegree (1, 1) on Γ\Gcan have maximal rank. QED

From now on, for any triple (Γ\G, J, ω) we shall assume that Γ\G is a compact 2n-dimensionalnilmanifold endowed with a NCS J having a compatible symplectic structure ω. We remarkthat ω might not come from a left invariant symplectic form on G.

The following theorem characterizes the compact nilmanifolds satisfying the Lefschetz com-plex condition of type (0, 1).

Theorem 3.3 (Γ\G, J, ω) has Lefschetz complex type (0, 1) if and only if Γ\G is a complextorus.

Proof : We shall prove that under this Lefschetz complex condition, the NCS J must becomplex parallelizable, and then the result follows from Theorem 3.2. In our proof we followLupton-Oprea approach [12].

Let us suppose that b1,0 6= n; then, without loss of generality we can suppose that xn1,0 in

Proposition 2.3 satisfies ∂(xn1,0) 6∈ 〈∂(x1

1,0), . . . , ∂(xn−11,0 )〉. This ensures that for each cohomology

class [z] ∈ H1,0(M), the generator xn1,0 does not appear in any representative of [z].

7

Let ω ∈ M1,1 be a cocycle representing the class of [ω] in H1,1(M). Then ω must beexpressed as

ω =∑

1≤i≤n−11≤j≤n

tij xi1,0x

j0,1 + xn

1,0y ,

where tij are complex numbers and y has bidegree (0,1). Notice that y must be nonzero, becauseotherwise ωn = 0. Now, since

∂ω =∑

1≤i≤n−11≤j≤n

tij(∂xi1,0)x

j0,1 −

∑

1≤i≤n−11≤j≤n

tij xi1,0(∂xj

0,1) + (∂xn1,0)y − xn

1,0∂y = 0,

it follows that xn1,0∂y = 0 because xn

1,0 cannot appear at any of the other summands, and hence∂y = 0. Thus y defines a nonzero cohomology class [y] ∈ H0,1(M).

Let us now consider the derivation θ = θn1,0:M∗,∗ −→ M∗−1,∗ defined in Section 2. From

Corollary 2.6 it induces a derivation θ: H∗,∗(M) −→ H∗−1,∗(M) on cohomology. Moreover,from θ(ω) = θ(xn

1,0)y − xn1,0θ(y) = y, we get that θ([ω]) = [y] 6= 0. But θ([z]) = 0, for any

[z] ∈ H1,0(M), and since the cup product [z][ωn] is above the maximum bidegree, it must bezero, and therefore

0 = θ([z][ωn]) = θ([z])[ωn]− [z]θ([ωn]),

i.e. [z]θ([ωn]) = 0 for any [z] ∈ H1,0(M). But θ([ωn]) ∈ Hn−1,n(M) and the pairing is non-singular (by Serre duality), hence we must have θ([ωn]) = 0, i.e.

0 = θ([ωn]) = nθ([ω])[ω]n−1 = nL0,1(θ([ω])).

Now the injectivity of L0,1 together with the fact that θ([ω]) = [y] implies [y] = 0, whichcontradicts the fact that [y] defines a nonzero cohomology class in H0,1(M).

Therefore, we conclude that if M has Lefschetz complex type (0, 1) then b1,0 = n, hence J

is complex parallelizable by Lemma 2.2, and the proof is complete. QED

Next, we study the injectivity of the homomorphism L1,0: H1,0(M) −→ Hn,n−1(M) given bythe cup product L1,0([α]) = [α][ω]n−1. This condition does not turn out to be so restrictive, butagain it characterizes an important class of complex nilmanifolds, namely those with abeliancomplex structure.

Theorem 3.4 Let (Γ\G, J, ω) be a compact 2n-dimensional nilmanifold as above. Then, it hasLefschetz complex type (1, 0) if and only if the complex structure J is abelian.

Proof : The fact that the injectivity of L1,0 implies that J must be an abelian complex structurecan be proved in a similar way as Theorem 3.3. Let us sketch the proof.

If b0,1 6= n, from Proposition 2.3 we can assume the minimal model with the generator xn0,1

satisfying ∂(xn0,1) 6∈ 〈∂(x1

0,1), . . . , ∂(xn−10,1 )〉. Thus xn

1,0 cannot appear in any representative of anycohomology class [z] ∈ H0,1(M).

Let ω ∈M1,1 be a cocycle representing the class [ω] in H1,1(M); then ω is expressed as

ω =∑

1≤i≤n1≤j≤n−1

tij xi1,0x

j0,1 + xn

0,1y ,

8

where tij are complex numbers and y has bidegree (1,0). Now, ∂ω = 0 implies that ∂(y) = 0and hence y defines a nonzero cohomology class [y] in H1,0(M).

From Corollary 2.6, the derivation θ = θn0,1 satisfies θ([ω]) = [y] 6= 0. Since θ([z]) = 0, for

any [z] ∈ H0,1(M), and [z][ωn] = 0, we get

0 = θ([z][ωn]) = θ([z])[ωn]− [z]θ([ωn]),

i.e. [z]θ([ωn]) = 0 for any [z] ∈ H0,1(M). But θ([ωn]) ∈ Hn,n−1(M) and by Serre duality wemust have θ([ωn]) = 0, i.e.

0 = θ([ωn]) = nθ([ω])[ω]n−1.

The injectivity of L1,0 implies that θ([ω]) = [y] = 0, which contradicts the fact that [y] definesa nonzero cohomology class in H1,0(M).

Therefore, one concludes that if L1,0 is injective then b0,1 = n, and the NCS J is abelian byLemma 2.2.

Let us show now that for any abelian NCS J the map L1,0 is a monomorphism, i.e. we mustshow that if all the Aijk in Proposition 2.3 vanish then L1,0 is injective.

It is clear that if J is abelian then ∂(Mn,r) = 0 for each r ≥ 0, so Hn,r(M) = Mn,r. SinceH1,0(M) = {z ∈ M1,0 | ∂z = 0} ⊂ M1,0, it suffices to prove that L1,0(z) = 0 implies z = 0.But this follows from Proposition 2.7 for r = n− 1. QED

According to Hodge theorem, on any compact Kahler manifold M , the Dolbeault cohomologygroups Hp,q(M) and Hq,p(M) are isomorphic by conjugation. From the previous theorem andSerre duality we get

Corollary 3.5 For any compact 2n-dimensional nilmanifold (Γ\G, J, ω) as above with J abelian,there is an injective homomorphism Hn−1,n

∂(Γ\G) ↪→ Hn,n−1

∂(Γ\G).

Moreover, from Theorem 3.4 it follows that any nilmanifold of Lefschetz complex type (0, 1)has also Lefschetz complex type (1, 0), but the converse is not true. In fact, we have:

Corollary 3.6 The map L1,0: H1,0(Γ\G) −→ Hn,n−1(Γ\G) is an isomorphism if and only ifΓ\G is a complex torus.

Therefore, either from Theorem 3.3 or Corollary 3.6, we conclude that a compact nilmanifoldwith NCS has no Kahler metrics unless it is a complex torus.

4 Dolbeault homotopy theoretic version of Hasegawa theorem

A DBAM = (M∗,∗, ∂) is said to be formal if there is a morphism ψ : (M∗,∗, ∂) −→ (H∗,∗(M), 0)of DBAs inducing the identity on cohomology. We shall say that a complex manifold M isDolbeault formal if there exists a minimal model M for (Λ∗,∗(M), ∂) which is formal.

Some examples of compact nilmanifolds with NCS whose Dolbeault minimal model is notformal are described in [6]; the non-formality of such particular examples followed from the exis-tence of nonzero Massey triple products. Our main goal in this section is to prove the followinggeneral characterization of the formality of the minimal model M given in Proposition 2.3.

9

Theorem 4.1 Let Γ\G be a compact nilmanifold with a NCS. Then, M is formal if and onlyif Γ\G is a complex torus.

Proof : First, suppose that b1,0 < n and M is formal, i.e. there exists a morphism of DBAsψ : (M∗,∗, ∂) −→ (H∗,∗(M), 0) inducing the identity on cohomology. Then ψ maps every ∂–closed element of M to its cohomology class. Let ψ1,0 be the restriction of ψ to M1,0. If{x1

0,1, . . . , xn0,1, x

11,0, . . . , x

n1,0} is a basis of M as given in Proposition 2.3, then {[x1

1,0], . . . , [xb1,0

1,0 ]}is a basis of H1,0(M), andM1,0 is generated by {x1

1,0, . . . , xb1,0

1,0 , yb1,0+11,0 , . . . , yn

1,0}, with ψ1,0(xi1,0) =

[xi1,0] ∈ H1,0(M) for i = 1, . . . , b1,0, and {yb1,0+1

1,0 , . . . , yn1,0} being a basis of kerψ1,0. Moreover,

from the definition of b1,0 it follows that there is no ∂–closed linear combination of the yi1,0’s.

Since there exists a nonzero scalar a such that x11,0 · · ·xn

1,0 = a x11,0 · · ·xb1,0

1,0 yb1,0+11,0 · · · yn

1,0, we getψ(x1

1,0 · · ·xn1,0) = 0. But x1

1,0 · · ·xn1,0 is ∂–closed, and therefore ψ(x1

1,0 · · ·xn1,0) = [x1

1,0 · · ·xn1,0] ∈

Hn,0(M). Hence [x11,0 · · ·xn

1,0] = 0, which contradicts Lemma 2.4. Hence, M is not formal ifb1,0 < n.

Next, suppose that b1,0 = n, b0,1 < n and M is formal. Let ψ0,1 denote the restrictionof ψ to M0,1. Again, as above, we know that {[x1

0,1], . . . , [xb0,1

0,1 ]} is a basis of H0,1(M), and

hence M0,1 is generated by x10,1, . . . , x

b0,1

0,1 and a basis {yb0,1+10,1 , . . . , yn

0,1} of kerψ0,1. Once more,

x10,1 · · ·xn

0,1 = b x10,1 · · ·xb0,1

0,1 yb0,1+10,1 · · · yn

0,1, for some nonzero scalar b, and hence ψ(x10,1 · · ·xn

0,1) =0. But ψ(x1

0,1 · · ·xn0,1) = [x1

0,1 · · ·xn0,1] ∈ H0,n(M), because x1

0,1 · · ·xn0,1 is ∂–closed, and we find

again a contradiction to Lemma 2.4.Therefore, M is not formal unless b1,0 = b0,1 = n. Thus, from Lemma 2.2 it follows that M

is formal if and only if g is an Abelian Lie algebra or, equivalently, Γ\G is a complex torus.QED

Massey products in H∗,∗(Γ\G) can be defined in the standard way (see [6] for details), andthe existence of a nonzero Massey product is an obstruction to Dolbeault formality. When thefirst Betti number b1(Γ\G) = 3, the lowest possible value of b1 for a nilmanifold admitting aNCS, an explicit nonzero Massey triple product can be constructed easily.

Proposition 4.2 Let Γ\G be a compact nilmanifold with a NCS. If b1(Γ\G) = 3 then theMassey product 〈x1

1,0, x11,0, x

10,1〉 is nonzero.

Proof : If b1 = 3 then B211 in Proposition 2.3 must be nonzero and the generators x11,0, x

10,1, x

21,0

of the minimal model M satisfy ∂x11,0 = ∂x1

0,1 = 0 and x11,0x

10,1 = ∂(x2

1,0/B211). Thus theMassey product 〈x1

1,0, x11,0, x

10,1〉 is represented by (1/B211)x1

1,0x21,0, whose cohomology class in

H2,0/x11,0H

1,0 is nonzero. QED

Next we discuss some consequences of Theorem 4.1.First, according to Neisendorfer and Taylor, a minimal model for the Dolbeault complex of

a compact Kahler manifold is formal. Thus, a compact nilmanifold with a NCS does not admitKahler structures unless it is a complex torus. Since there exist compact non-toral nilmanifoldsΓ\G with NCS having compatible symplectic structures (see examples in Section 5), it followsthat Dolbeault formality provides a difference between indefinite and positive definite Kahlermanifolds.

10

In [14] the notion of strict formality is also introduced. Compact Kahler manifolds are strictlyformal, and strictly formal complex manifolds are in particular Dolbeault formal. However,Calabi–Eckmann manifolds are all Dolbeault formal, but not all of them are strictly formal [14].In this context, Theorem 4.1 implies:

Corollary 4.3 A compact nilmanifold with a NCS is strictly formal if and only if it is Dolbeaultformal.

Since a compact nilmanifold is de Rham formal if and only if it is a torus [11], we obtain:

Corollary 4.4 Let Γ\G be a compact nilmanifold with a NCS. The following conditions areequivalent:

(i) Γ\G is de Rham formal,

(ii) Γ\G is Dolbeault formal,

(iii) Γ\G is strictly formal,

(iv) Γ\G is a complex torus.

Remark 4.5 Tralle and Oprea raised the following question in [21, p.193]: are there complexsymplectic de Rham formal manifolds whose non-Kahlerness follows from the Dolbeault homo-topy theory? Corollary 4.4 shows that the answer to this question is negative in the family ofcompact nilmanifolds with NCS.

In relation to the degeneration of the Frolicher’s spectral sequence {Er}r≥1 associated to acomplex manifold [10], Tanre [19] has proved that if K is a compact connected semisimple Liegroup such that T ↪→ K → K/T is a holomorphic principal bundle, T being a maximal torus ofK, then E2(K) 6∼= E∞(K) if and only if K is not Dolbeault formal. These conditions are satisfiedby Pittie’s example [16], where K = SO(9). In contrast to Tanre’s result, and in spite of the factthat any Γ\G with NCS can be realized as the total space at the top of a tower of holomorphicprincipal bundles with complex tori as structure groups [6], for compact nilmanifolds with NCSwe have:

Corollary 4.6 Let Γ\G be a compact nilmanifold with a NCS. If E1(Γ\G) 6∼= E∞(Γ\G) thenΓ\G is not Dolbeault formal.

Proof : It follows immediately from Theorem 4.1, taking into account that any compact Kahlermanifold satisfies E1

∼= E∞ by Hodge theory. QED

However, the converse of this result does not hold in general. The manifold Γn\Gn describedin Example 5.1 below is a 2n-dimensional example whose Dolbeault minimal model is not formalbut with degenerate Frolicher spectral sequence.

5 Compact nilmanifolds of Lefschetz complex type (1, 0)

In this section we show the existence of large families of nilmanifolds admitting abelian complexstructures with compatible symplectic form, to which the characterization Theorem 3.4 applies.

11

5.1 A countable family of examples

Let us begin by recalling that the simplest non-trivial example of an abelian NCS is providedby the well-known Kodaira-Thurston manifold, which can be described as follows. Consider thereal 4-dimensional nilpotent Lie algebra g = 〈X1, . . . , X4 | [X1, X2] = X4〉; the correspondingsimply-connected Lie group G has a discrete subgroup Γ such that Γ\G is compact [13]. (TheKodaira-Thurston manifold KT = Γ\G was the first known example of a compact symplectic(and complex) manifold with no positive definite Kahler metric [20].) Any complex structure J

on g is abelian [18], so for any J there is a basis {α1, α2, α1, α2} of (gC)∗ such that

dα1 = 0, dα2 = α1 ∧ α1.

Moreover, any such J has a compatible symplectic structure ω, namely ω =√−1 (α1 ∧ α2 +

α2 ∧ α1), and the corresponding (indefinite) metric is given by ds2 = α1#α2 + α2#α1, where #denotes the symmetric product.

Next, we consider non-trivial extensions of KT given as follows. Let gCn be the complexi-fication of the real 2n-dimensional nilpotent Lie algebra gn characterized by the existence of abasis {α1, . . . , αn, α1, . . . , αn} of the dual (gCn )∗ satisfying

dα1 = 0, dαj =j−1∑

k=1

αk ∧ αj−k, j = 2, . . . , n,(3)

where the αj ’s are assumed to have type (1,0) and the αj ’s are their complex conjugate, andhence of type (0, 1). By definition, gn has an abelian NCS and, since the corresponding simply-connected Lie group Gn has a discrete co-compact subgroup Γn by [13], the compact nilmanifoldΓn\Gn has an abelian complex structure. Notice that Γ2\G2 is precisely the Kodaira-Thurstonmanifold KT .

Let ω be the real (1, 1)-form on Γn\Gn given by

ω =√−1

n∑

k=1

αk ∧ αn+1−k.(4)

It is straightforward to check that ω is closed and of maximal rank. Thus ω is a symplecticstructure on Γn\Gn compatible with its abelian complex structure.

Thus, in contrast with Theorem 3.2, this family of examples shows that:

Theorem 5.1 For each n ≥ 1, there is a 2n-dimensional compact nilmanifold Γn\Gn admittingan abelian complex structure that has compatible symplectic forms.

It is worthy to remark that for each n the structure equations defining Γn+1\Gn+1 are givenby adding the equation dαn+1 = −√−1ω to (3), where ω is the symplectic form (4) considered onΓn\Gn. Moreover, each manifold Γn\Gn is the total space of a holomorphic principal bundle [6]

T1 ↪→ Γn\Gn −→ Γn−1\Gn−1

with structure group the complex torus T1 and base Γn−1\Gn−1.Since the computation of the terms Ep,q

r (Γn\Gn) in the Frolicher spectral sequence reduces tothe Lie algebra level [4], a long but straightforward computation shows that there is a Hodge type

12

decomposition for Γn\Gn in the sense that Hk(Γn\Gn) ∼= ⊕p+q=kHp,q(Γn\Gn), i.e. the Frolicher

spectral sequence degenerates at the first level. However, as it is shown by the Example 5.4 below,there exist abelian NCSs for which their associated sequence satisfies E2 6∼= E∞.

5.2 Two examples in dimension six

Now we restrict our attention to six dimensional compact nilmanifolds Γ\G. Equations (2)imply that the Lie algebra g must be nilpotent in step s ≤ 3. If s = 1 then the Lie algebra isAbelian, so we consider next two interesting examples, for s = 2, 3, in which any NCS is abelianand admits a compatible symplectic structure.

Proposition 5.2 Let Γ\G be a 6-dimensional compact nilmanifold corresponding to the 2-stepnilpotent Lie algebra g given by

g = 〈X1, . . . , X6 | [X1, X2] = X6〉.

Then, any complex structure J on Γ\G, coming from a left invariant one, is abelian and admitsa compatible form ω.

Proof : Let us prove first that, since Γ\G has first Betti number equal to 5, any complexstructure J on Γ\G coming from a left invariant one must be abelian. In fact, if there is anelement α ∈ Λ1,0(gC)∗ whose differential has nonzero component of bidegree (2,0) then dα hasnonzero (0,2)-component, so the first Betti number of Γ\G would be ≤ 4. Thus, d(Λ1,0) ⊂ Λ1,1

and J is abelian. Therefore, its complex structure equations are of the form

dα1 = dα2 = 0, dα3 = Q α1 ∧ α1 + R α1 ∧ α2 + S α2 ∧ α1 + T α2 ∧ α2,

where R = S ∈ C and Q,T ∈ R. Moreover, since the second Betti number is 11, these coefficientsmust satisfy QT = |R|2. Now, if R = T = 0 then the real 2-form ω =

√−1 (α1 ∧ α3 + α2 ∧ α2 +α3 ∧ α1) is closed and nondegenerate. Otherwise, i.e. if R and T do not vanish simultaneously,then ω =

√−1α1∧ α1 +√−1α2∧ α2 +R ,α1∧ α3 +T α2∧ α3− R α3∧ α1−T lpha3∧ α2 is real,

closed and nondegenerate, because ω3 = −6(|R|2 + T 2)√−1α1 ∧ α2 ∧ α3 ∧ α1 ∧ α2 ∧ α3 6= 0. In

both cases, the symplectic form ω has bidegree (1,1) and so it is compatible with J . QED

Proposition 5.3 Let Γ\G be a 6-dimensional compact nilmanifold corresponding to the 3-stepnilpotent Lie algebra g given by

g = 〈X1, . . . , X6 | [X1, X2] = X5, [X1, X4] = [X2, X5] = X6〉.

Then, any NCS on Γ\G is abelian and admits compatible symplectic structures.

Proof : The complex structure equations of any NCS J on Γ\G are of the form

dα1 = 0, dα2 = λα1 ∧ α1, dα3 = P α1 ∧ α2 + Qα1 ∧ α1 + R α1 ∧ α2 + S α2 ∧ α1 + T α2 ∧ α2,

where λ, P, Q,R, S, T ∈ C. Since g is 3-step nilpotent, the coefficient λ is necessarily nonzero,so we can suppose λ = 1 and Q = 0. Thus, P must be zero, because the first Betti number isequal to 4 and therefore J is abelian.

13

Moreover, the Jacobi identity for the Lie algebra, which is equivalent to d2 = 0, implies thatT = 0. Therefore, the complex equations of J are reduced to

dα1 = 0, dα2 = α1 ∧ α1, dα3 = R α1 ∧ α2 + S α2 ∧ α1,

where the complex coefficients R, S 6= 0, because otherwise the first Betti number would not beequal to 4. Thus, the form ω =

√−1 ((1/R)α1 ∧ α3 + α2 ∧ α2 + (1/R)α3 ∧ α1) is a symplecticform compatible with J . QED

5.3 A 1-parametric family of abelian complex structures on the Iwasawa

manifold

The Iwasawa manifold can be realized as the quotient of the complex nilpotent Lie group

G =

1 x z

0 1 y

0 0 1

| x, y, z ∈ C

by the subgroup Γ ⊂ G consisting of those matrices whose entries {x, y, z} are Gaussian integers.The complex differential forms dx, dy and dz − xdy on G are left invariant and descend toholomorphic 1–forms β1, β2 and β3 on Γ\G satisfying

dβ1 = dβ2 = 0, dβ3 = −β1 ∧ β2.

The Iwasawa manifold Γ\G is, apart from a torus, the simplest example of a compact complexparallelizable nilmanifold. According to Proposition 3.2, no compatible symplectic structure canexist on Γ\G. However, there exist abelian NCSs on the manifold I underlying the Iwasawamanifold that do posses compatible symplectic forms. In fact, let {X1, Y1, X2, Y2, X3, Y3} bethe basis dual to {τ1, ν1, τ2, ν2, τ3, ν3}, where τi = Re (βi), νi = Im (βi). In [9] it is defined thealmost complex structure Jt given by

JtX1 = cos tX2 + sin t Y2, JtY1 = − sin t X2 + cos t Y2, JtX3 = Y3,

for each t ∈ R. It is easy to see that there is a basis {α1, α2, α3} of complex forms of bidegree(1,0) with respect to Jt, for which the structure equations become

dα1 = dα2 = 0, dα3 = λ (α1 ∧ α2 + α2 ∧ α1) + µ (α1 ∧ α1 − α2 ∧ α2),(5)

where λ = cos t +√−1 sin t and µ = sin t − √−1 cos t. These equations prove that Jt is an

abelian NCS for each t ∈ R. Moreover, a symplectic form compatible with Jt is given by

ωt = λ α2 ∧ α3 + λ α2 ∧ α3 + µ α1 ∧ α3 + µ α1 ∧ α3 + α1 ∧ α2 + α1 ∧ α2.

Let us remark that 〈α1, α1 ∧ α2, µ α1 + λ α2〉 defines a nonzero triple Massey product on(I, Jt), for each t ∈ R. Also, a direct computation shows that the Frolicher spectral sequenceassociated to Jt satisfies E1 6∼= E2

∼= E∞, for each t ∈ R; more precisely,

dimE1(I, Jt) = 48 > dimE2(I, Jt) = dimE∞(I, Jt) = 36.

14

Finally, by integrating the structure equations (5) one can realize (I, Jt) as the quotient of

Gt =

1 λu− µv µu + λv w

0 1 0 v

0 0 1 u

0 0 0 1

| u, v, w ∈ C

by the subgroup Γt consisting of those matrices whose entries {u, v, w} are Gaussian integers.

5.4 A 3-parametric family of examples in dimension six

For each triple (A,B,C) of rational numbers, let gABC be the real 6-dimensional nilpotent Liealgebra underlying gCABC given by

dα1 = 0, dα2 = α1 ∧ α1, dα3 = Aα1 ∧ α2 + B α1 ∧ α2 + C α2 ∧ α1,(6)

where {α1, α2, α3, α1, α2, α3} is a basis of the dual (gCABC)∗. Since (A,B,C) ∈ Q3, the cor-responding nilpotent Lie group GABC admits a discrete co-compact subgroup ΓABC (see [5]for an explicit description of GABC). Therefore, ΓABC\GABC is a compact nilmanifold with aNCS. These manifolds can be described as the total space of a holomorphic principal bundleT1 ↪→ ΓABC\GABC −→ KT with structure group the complex torus T1 and base the Kodaira-Thurston manifold.

Proposition 5.4 [5] Let us suppose (A,B, C) ∈ Q3 − {(0, 0, 0)}. Then:

(i) ΓABC\GABC has symplectic structure compatible with its complex structure iff A+B 6= 0.

(ii) If A+B = 0 then, the manifold underlying ΓABC\GABC has symplectic structure iff C 6= 0.

In particular, if A = 0 then the abelian NCS given by (6) has no compatible symplecticstructure (homogeneous or otherwise) if and only if B = 0, although the manifold underlyingΓABC\GABC has symplectic structure because C must be nonzero.

If A + B 6= 0 then ds2 = α1#α3 + (A + B) α2#α2 + α3#α1 is an indefinite Kahler metric onΓABC\GABC , and its corresponding Kahler form is the symplectic form

ω =√−1 (α1 ∧ α3 + (A + B) α2 ∧ α2 + α3 ∧ α1).

Notice that, since ∂(α1 ∧ α2 ∧ α3 ∧ α3) = −Aα1 ∧ α2 ∧ α3 ∧ α1 ∧ α2, the homomorphism

L1,0:H1,0(ΓABC\GABC) −→ H3,2(ΓABC\GABC)

satisfies that L1,0(α1) = α1 ∧ [ω]2 = −2(A + B)[α1 ∧ α2 ∧ α3 ∧ α1 ∧ α2] is nonzero if and only ifA = 0, i.e. the NCS is abelian, according to Theorem 3.4.

In relation to the Frolicher’s spectral sequence, the following table illustrates the dimensionof each term Er(Γ0BC\G0BC), for any manifold Γ0BC\G0BC in the family for which the NCS is

15

abelian:

Parameters B, C dimE1 dimE2 dimE3 Sequence {Er}

B 6= 0 ; C = 0 40 24 24 E1 6∼=E2∼=E∞

B = 0 ; C 6= 0 32 28 24 E1 6∼=E2 6∼=E3∼=E∞

B,C 6=0C2 6= B2 32 28 24 E1 6∼=E2 6∼=E3

∼=E∞

C2 = B2 32 32 32 E1∼=E∞

Acknowledgments.- This work has been partially supported through grants DGICYT (Spain)Projects PB-97-0504-C02-01/02 and UPV Project 127.310-EA 043/97.

References

[1] M.L. Barberis, I.G. Dotti Miatello, R.J. Miatello, On certain locally homogeneous Clifford manifolds,Ann. Glob. Anal. Geom. 13 (1995), 289–301.

[2] C. Benson, C.S. Gordon, Kahler and symplectic structures on nilmanifolds, Topology 27 (1988),513–518.

[3] L.A. Cordero, M. Fernandez, A. Gray, Symplectic manifolds with no Kahler structure, Topology 25(1986), 375–380.

[4] L.A. Cordero, M. Fernandez, A. Gray, L. Ugarte, Frolicher spectral sequence of compact nilmanifoldswith nilpotent complex structure, New Developments in Differential Geometry, Budapest 1996 (J.Szenthe Ed.), Proc. Conf. on Diff. Geometry , Budapest, July 27-30, 1996; p. 77–102. Kluwer Acad.Publ., Dordrecht 1998.

[5] L.A. Cordero, M. Fernandez, A. Gray, L. Ugarte, Dolbeault homotopy theory and compact nilman-ifolds, Homotopy and Geometry (J. Oprea and A. Tralle Eds.), Proc. Workshop on Homotopy and

Geometry, June 9–13, 1997, Warszawa (Poland), Banach Center Publ. 45 (1998), 137–154.

[6] L.A. Cordero, M. Fernandez, A. Gray, L. Ugarte, Compact nilmanifolds with nilpotent complexstructure: Dolbeault cohomology, Trans. Amer. Math. Soc. 352 (2000), 5405–5433.

[7] P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kahler manifolds, Invent.Math. 29 (1975), 245–274.

[8] I.G. Dotti Miatello, A. Fino, Abelian hypercomplex 8-dimensional nilmanifolds, Ann. Glob. Anal.Geom. 18 (2000), 47–59.

[9] M. Fernandez, A. Gray, The Iwasawa manifold. Differential geometry, Penıscola 1985, 157–159,Lecture Notes in Math. 1209, Springer, Berlin-New York, 1986.

[10] A. Frolicher, Relations between the cohomology groups of Dolbeault and topological invariants, Proc.Nat. Acad. Sci. U.S.A. 41 (1955), 641–644.

[11] K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), 65–71.

[12] G. Lupton, J. Oprea, Symplectic manifolds and formality, J. Pure Appl. Algebra 91 (1994), 193–207.

[13] I.A. Mal’cev, A class of homogeneous spaces, Amer. Math. Soc. Transl. No. 39 (1951).

16

[14] J. Neisendorfer, L. Taylor, Dolbeault Homotopy Theory, Trans. Amer. Math. Soc. 245 (1978), 183–210.

[15] K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. ofMath. 59 (1954), 531–538.

[16] H. Pittie, The nondegeneration of the Hodge-de Rham spectral sequence, Bull. Amer. Math. Soc.20 (1989), 19–22.

[17] Y. Sakane, On compact parallelizable solvmanifolds, Osaka J. Math. 13 (1976), 187–212.

[18] S. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311–333.

[19] D. Tanre, Modele de Dolbeault et fibre holomorphe, J. Pure Appl. Algebra 91 (1994), 333–345.

[20] W.P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976),467–468.

[21] A. Tralle, J. Oprea, Symplectic manifolds with no Kahler structure, Lecture Notes in Math. 1661,Springer (1997).

[22] H.C. Wang, Complex parallelisable manifolds, Proc. Amer. Math. Soc. 5 (1954), 771–776.

[23] R.O. Wells, Differential Analysis on Complex Manifolds, Graduate Texts in Math. 65, Springer(1980).

L.A. Cordero: Departamento de Geometrıa y Topologıa, Facultad de Matematicas, Universidad de San-tiago de Compostela, 15706 Santiago de Compostela, Spain. E-mail: cordero@zmat.usc.es

M. Fernandez: Departamento de Matematicas, Facultad de Ciencias, Universidad del Paıs Vasco, Apartado644, 48080 Bilbao, Spain. E-mail: mtpferol@lg.ehu.es

L. Ugarte: Departamento de Matematicas, Facultad de Ciencias, Universidad de Zaragoza, Campus PlazaSan Francisco, 50009 Zaragoza, Spain. E-mail: ugarte@posta.unizar.es

17