© Hindawi Publishing Corp. COHOMOLOGICALLY KÄHLER MANIFOLDS WITH NO KÄHLER METRICS

We show some examples of compact symplectic solvmanifolds, of 
dimension greater than four, which are cohomologically 
Kahler and do not admit Kahler metric since their 
fundamental groups cannot be the fundamental group of any compact 
Kahler manifold. Some of the examples that we study were 
considered by Benson and Gordon (1990). However, whether such 
manifolds have Kahler metrics was an open question. The 
formality and the hard Lefschetz property are studied for the 
symplectic submanifolds constructed by Auroux (1997) and some 
consequences are discussed.


Introduction.
A symplectic manifold (M, ω) is a pair consisting of a 2ndimensional differentiable manifold M together with a closed 2-form ω which is nondegenerate (i.e., ω n never vanishes).The form ω is called symplectic.By the Darboux theorem, in canonical coordinates, ω can be expressed as (1.1) Any symplectic manifold (M, ω) carries an almost complex structure J compatible with the symplectic form ω, which means that ω(X, Y ) = ω(JX, JY ) for any X, Y vector fields on M (see [22,23]).If (M, ω) has an integrable almost complex structure J compatible with the symplectic form ω such that the Riemannian metric g, given by g(X, Y ) = −ω(JX, Y ), is positive definite, then (M,ω,J) is said to be a Kähler manifold with Kähler metric g.
The problem of how compact symplectic manifolds differ topologically from Kähler manifolds led, during the last years, to the introduction of several geometric methods for constructing symplectic manifolds (see [5,8,15,20,21]).The symplectic manifolds presented there do not admit a Kähler metric since they are not formal or do not satisfy hard Lefschetz theorem, or they fail both properties of compact Kähler manifolds.
The purpose of this paper is to show that the formality and the hard Lefschetz property of any compact symplectic manifold M are not sufficient conditions to imply the existence of a Kähler metric on M. We describe three families of compact symplectic solvmanifolds M 6 (c), P 6 (c), and N 6 (c) of dimension 6, and a family of compact symplectic solvmanifolds N 8 (c) of dimension 8, each of which is formal and satisfies the hard Lefschetz property.Thus, they are cohomologically Kähler, their odd Betti numbers are even (see [19]), and their even Betti numbers are nonzero.
In [13], there are given examples of 4-dimensional compact symplectic manifolds which are cohomologically Kähler but do not possess complex structures, so they admit no Kähler metrics.This is done by appealing to classification theorems of Kodaira and Yau that are specific to complex dimension 2.
In our case, we resort, in Section 3, to the properties of the fundamental group of a compact Kähler manifold given by Campana [7] to show that none of the manifolds M 6 (c), N 6 (c), P 6 (c), and N 8 (c) admit Kähler metrics (see Theorems 3.3 and 3.5).A similar technique was used in [14] to prove the existence of 4-dimensional Donaldson symplectic submanifolds with no complex structures.The manifolds N 6 (c) as well as the manifolds P 6 (c) were considered in [6].There, Benson and Gordon show that they are cohomologically Kähler.However, whether or not they have a Kähler metric was an open question.
On the other hand, in Section 4, we study the formality and the hard Lefschetz property for the symplectic submanifolds obtained by Auroux in [3] as an extension to higher-rank bundles of the symplectic submanifolds constructed by Donaldson in [11].Let (M, ω) be a compact symplectic manifold of dimension 2n with [ω] ∈ H 2 (M) having a lift to an integral cohomology class, and let E be any Hermitian vector bundle over M of rank r .In [3], Auroux proved the existence of some integer number k 0 such that for any k ≥ k 0 , there is a symplectic submanifold Z r M of dimension 2(n−r ) whose homology class realizes the Poincaré dual of where c i (E) denotes the ith Chern class of the vector bundle E. For such manifolds the inclusion j : Z r M induces on cohomology: (i) an isomorphism j * : As a consequence of this study, we get some examples of Auroux symplectic submanifolds (in particular, nonparallelizable manifolds) of dimension 6 which are formal and hard Lefschetz, but do not carry Kähler metrics.
2. Formal manifolds.First, we need some definitions and results about minimal models.Let (A, d) be a differential algebra, that is, A is a graded commutative algebra over the real numbers, with a differential d which is a derivation, that is, , where deg(a) is the degree of a.
A differential algebra (A, d) is said to be minimal if (i) A is free as an algebra, that is, A is the free algebra V over a graded vector space V = ⊕V i , (ii) there exists a collection of generators {a τ , τ ∈ I}, for some well-ordered index set I, such that deg(a µ ) ≤ deg(a τ ) if µ < τ and each da τ is expressed in terms of preceding a µ (µ < τ).This implies that da τ does not have a linear part, that is, it lives in Morphisms between differential algebras are required to be degree-preserving algebra maps which commute with the differentials.Given a differential algebra (A, d), we denote by H * (A) its cohomology.We say that A is connected if H 0 (A) = R, and A is one-connected if, in addition, H 1 (A) = 0.
We will say that (ᏹ,d) is a minimal model of the differential algebra (A, d) if (ᏹ,d) is minimal and there exists a morphism of differential graded algebras ρ : (ᏹ,d) → (A, d) inducing an isomorphism ρ * : H * (ᏹ) → H * (A) on cohomology.Halperin [17] proved that any connected differential algebra (A, d) has a minimal model unique up to isomorphism.
A minimal model (ᏹ,d) is said to be formal if there is a morphism of differential algebras ψ : (ᏹ,d) → (H * (ᏹ), d = 0) that induces the identity on cohomology.The formality of a minimal model can be distinguished as follows.
Theorem 2.1 (see [10]).A minimal model (ᏹ,d) is formal if and only if ᏹ = V and the space V decomposes as a direct sum A minimal model of a connected differentiable manifold M is a minimal model ( V ,d) for the de Rham complex (ΩM, d) of differential forms on M. If M is a simply connected manifold, the dual of the real homotopy vector space π i (M) ⊗ R is isomorphic to V i for any i.We will say that M is formal if its minimal model is formal or, equivalently, the differential algebras (ΩM, d) and (H * (M), d = 0) have the same minimal model.(For details see, for example, [10,16]) In [14], the condition of formal manifold is weaken to s-formal manifold as follows.
Definition 2.2.Let (ᏹ,d) be a minimal model of a differentiable manifold M. We say that (ᏹ,d) is s-formal, or M is an s-formal manifold (s ≥ 0) if ᏹ = V such that for each i ≤ s, the space V i of generators of degree i decomposes as a direct sum V i = C i ⊕ N i , where the spaces C i and N i satisfy the three following conditions: The relation between the formality and the s-formality for a manifold is given in the following theorem.
Theorem 2.3 (see [14]).Let M be a connected and orientable compact differentiable manifold of dimension 2n or (2n − 1).Then M is formal if and only if it is (n − 1)-formal.

Formal and hard Lefschetz symplectic manifolds with no Kähler metric.
In this section, we show the existence of compact symplectic manifolds of dimension greater than 4, which do not admit Kähler metrics even when they are formal and hard Lefschetz.
Example 3.1 (the manifolds M 6 (c) [9]).Let G(c) be the connected completely solvable Lie group of dimension 5 consisting of matrices of the form where x i ,y i ,z ∈ R (i = 1, 2) and c is a nonzero real number.Then a global system of coordinates x 1 , y 1 , x 2 , y 2 , and z for G(c) is given by x i (a) = x i , y i (a) = y i , and z(a) = z.A standard calculation shows that a basis for the right invariant 1-forms on G(c) consists of Alternatively, the Lie group G(c) may be described as a semidirect product , where ψ(z) is the linear transformation of R 4 given by the matrix for any z ∈ R. Thus, G(c) has a discrete subgroup Γ (c) = Z ψ Z 4 such that the quotient space Γ (c)\G(c) is compact.Therefore, the forms dx i − cx i dz, dy i + cy i dz, and dz (i = 1, 2) descend to 1-forms α i , β i , and Hence, there are 1-forms α 1 , β 1 , α 2 , β 2 , γ, and η on M 6 (c) such that where i = 1, 2, and such that at each point of M 6 (c), {α 1 ,β 1 ,α 2 ,β 2 ,γ,η} is a basis for the 1-forms on M 6 (c).Using Hattori's theorem [18], we compute the real cohomology of M 6 (c): Therefore, the Betti numbers of M 6 (c) are Proposition 3.2.The manifold M 6 (c) is 2-formal and so formal.Moreover, M 6 (c) has a symplectic form ω such that (M 6 (c), ω) satisfies the hard Lefschetz property.
Proof.To prove that M 6 (c) is 2-formal, we see that its minimal model must be a differential graded algebra (ᏹ,d), ᏹ is the free algebra of the form where the generators a i have degree 1, the generators b j have degree 2, and the differential d is given by da i = db j = 0, where i = 1, 2 and 1 ≤ j ≤ 4. The morphism ρ : ᏹ → Ω(M), inducing an isomorphism on cohomology, is defined by ρ(a According to Definition 2.2, we get 4 and N 2 = 0. Now, the formality of M 6 (c) follows from Theorem 2.3.
The manifolds M 6 (c) were considered in [9].There, the formality of M 6 (c) is obtained as a consequence of the existence of a morphism (H * (M 6 (c)), d = 0) → (Ω * (M 6 (c)), d) that induces the identity on cohomology.Such a morphism is defined by linearity choosing closed forms representatives for each cohomology class.However, whether or not M 6 (c) has a Kähler metric was an open question.
Proof.In order to show that M 6 (c) does not admit Kähler metric, notice that Γ = π 1 (M 6 (c)) is a product Γ = Γ (c) × Z.Moreover, its abelianization is H 1 (M 6 (c); Z), and thus, it has rank 2. We will see that Γ cannot be the fundamental group of any compact Kähler manifold.
The exact sequence shows that Γ is solvable of class 2, that is, D 3 Γ = 0.Moreover, its rank is 6 by additivity (see [1] for details).Assume now that Γ = π 1 (X), where X is a compact Kähler manifold.According to Arapura-Nori's theorem (see [2, Theorem 3.3]), there exists a chain of normal subgroups such that Q is torsion, P /Q is nilpotent, and Γ /P is finite.The exact sequence (3.8) implies that Γ has no torsion, and so Q = 0.As Γ /P is torsion, thus finite, we have rank P = rank Γ = 6.Now, the finite inclusion P ⊂ Γ defines a finite cover p : Y → X that is also compact Kähler and it has fundamental group P .We show that P cannot be the fundamental group of any compact Kähler manifold.For this, we use Campana's result (see [7,Corollary 3.8,page 313]) that states that if G is the fundamental group of a Kähler manifold such that G is nilpotent and non-abelian, then G has rank greater than or equal to 9.
Since P is the fundamental group of the Kähler manifold Y , P is nilpotent, it has rank less than 9, and it has to be abelian.This is impossible since any pair of nonzero elements e ∈ Z 2 ⊂ Γ = Z 2 Z 4 , f ∈ Z 4 ⊂ Γ do not commute (see, e.g., [12, page 22]).
Example 3.4 (the manifolds N 6 (c)).We consider the connected completely solvable Lie group G(c) of dimension 3 consisting of matrices of the form We use again Hattori's theorem [18] to compute the real cohomology of Sol( 3) Denote by M 4 (c) the product M 4 (c) = Sol(3) × S 1 .In [13], it is proved that M 4 (c) is cohomologically Kähler (in fact, it has the same minimal model as T 2 × S 2 ) and it does not carry complex structures, and so it carries no Kähler metrics.This is done by appealing to classification theorems of Kodaira and Yau that are specific to complex surfaces.
It is clear that N 8 (c) is a symplectic manifold since it is the product of symplectic manifolds.In fact, a symplectic form ω 3 on N 8 (c) is given by where η is a symplectic form on the 2-torus T 2 .
One can check that the manifolds N 6 (c), P 6 (c), and N 8 (c) are cohomologically Kähler.Now, using an argument similar to the one given in Theorem 3.3, we get the following theorem.
We notice that the manifolds N 6 (c) and P 6 (c) were considered as examples of cohomologically Kähler manifolds by Benson and Gordon in [6].However, whether or not they have a Kähler metric was an open question.

Formality and hard Lefschetz property for Auroux symplectic submanifolds.
In this section, we study the conditions under which Auroux symplectic manifolds are formal and/or satisfy the hard Lefschetz theorem.
Let (M, ω) be a compact symplectic manifold of dimension 2n with [ω] ∈ H 2 (M) admitting a lift to an integral cohomology class, and let E be any Hermitian vector bundle over M of rank r .In [3] (E) for any integer number k large enough, where c i (E) denotes the ith Chern class of the vector bundle E.Moreover, these submanifolds satisfy a Lefschetz theorem in hyperplane sections, meaning that the inclusion j : Z r M is (n − r )-connected, that is, the map there j * : H i (M) → H i (Z r ) is an isomorphism for i < n − r and a monomorphism for i = n − r .
In general, let X and Y be compact manifolds.We say that a differentiable map f : X → Y is a homotopy s-equivalence (s ≥ 0) if it induces isomorphisms f * : H i (Y ) → H i (X) on cohomology for i < s, and a monomorphism f * : H s (Y ) H s (X) for i = s.Therefore, for any Auroux symplectic submanifold, the inclusion j : Z r M is a homotopy (n − r )-equivalence.
Theorem 4.1 (see [14]).Let X and Y be compact manifolds and let f : As a consequence of Theorem 4.1, we get the following corollary.
In order to continue the analysis of the Auroux symplectic submanifolds we introduce the following definition.Definition 4.3.Let (M, ω) be a compact symplectic manifold of dimension 2n.We say that M is s-Lefschetz with s ≤ (n − 1) if Proof.From now on, we denote by L the complex line bundle over M whose first Chern class is c 1 , and we consider the map j * : H p (M) → H p (Z r ) induced by the inclusion j on cohomology.First, we claim that for [z] ∈ H p (M) it holds that for large values of the parameter k.This can be shown via Poincaré duality.Clearly, j * [z] = 0 if and only if j * [z] • a = 0 for any a ∈ H i (Z r ).Since there is an isomorphism H i (Z r ) H i (M) for i ≤ (n − r − 1), we can assume that there exists a closed i-form x on M with [x| Zr ] = [ x] = a, x being the differential form on Z r given by x = j * (x).So .10)   where x, y, z ∈ R (i = 1, 2) and c is a nonzero real number.Then a global system of coordinates x, y, and z for G(c) is given by x(a) = x, y(a) = y, and z = z.A standard calculation shows that a basis for the right invariant 1-forms on G(c) consists of dx − cxdz, dy + cydz, dz .(3.11) Let Γ (c) be a discrete subgroup of G(c) such that the quotient space Sol(3) = Γ (c)\G(c) is compact (for the existence of such a subgroup Γ (c) see [4, page 20]).Hence, the forms dx − cxdz, dy + cydz, and dz all descend to 1-forms α, β, and γ on Sol(3) such that dα = −cα ∧ γ, dβ = cβ ∧ γ, dγ = 0. (3.12)

Corollary 4 . 2 .
Let M be a compact symplectic manifold of dimension 2n and let Z r M be an Auroux submanifold of dimension 2(n − r ).For each s
.1) is an isomorphism for all i ≤ s.By extension, if we say that M is s-Lefschetz with s ≥ n, then we just mean that M is hard Lefschetz.Let (M, ω) be a compact symplectic manifold of dimension 2n such that the de Rham cohomology class [ω] ∈ H 2 (M) has a lift to an integral cohomology class, and let Z r M be an Auroux submanifold of dimension 2(n− r ).Then, for large enough k and for each s