Coeffective cohomology of symplectic aspherical manifolds

We prove a generalization of the theorem which is proved by Fernandez, Ibanez, and de Leon. By this result, we give examples of non-K\"ahler manifolds which satisfy the property of compact K\"ahler manifolds concerning the coeffective cohomology.

Proposition 2.2. Let A * ⊂ A * (M ) be a sub-complex such that the inclusion Φ : A * → A * (M ) induces a cohomology isomorphism. Assume ω ∈ A * and the map ω∧ : A p → A p+2 is surjective for p ≥ n − 1. Denote A * coE = ker(ω∧) |A * . Then the inclusion Φ : A * coE → A * coE (M ) induces an isomorphism Proof. As above, we have the exact sequence of cochain complex By the assumption, for p ≥ n we have the long exact sequence of cohomology By the assumption Φ * : H * (A * ) → H * (A * (M )) is an isomorphism and so by this diagram Φ * : ) is an isomorphism.

Background: Fernandez-Ibanez-de Leon's theorem
Let G be a simply connected Lie group with a lattice (i.e. a cocompact discrete subgroup of G) Γ. We call G/Γ a nilmanifold (resp. solvmanifold) if G is nilpotent (resp. solvable). Let g be the Lie algebra of G and g * be the cochain complex of g with the differential which is induced by the dual of the Lie bracket. As we regard g * as the left-invariant forms on G/Γ, we consider the inclusion g * ⊂ A * (G/Γ). Let ω ∈ 2 g * be a left-invariant symplectic form. Then the map ω∧ : p g * → p+2 g * is surjective for p ≥ n−1(see [5]). In [14] Nomizu showed that if G is nilpotent then the inclusion g * ⊂ A * (G/Γ) induces an isomorphism of cohomology. Hence by Proposition 2.2, we have the following theorem which was noted in [5] and [6].
Theorem 3.1. Let G be a simply connected nilpotent Lie group with a lattice Γ and a left-invariant symplectic form ω. Then the inclusion g * ⊂ A * (G/Γ) induces an isomorphism for p ≥ n where coE g * = {α ∈ g * |ω ∧ α = 0}.
In [8] Hattori showed that the isomorphism H * ( g * ) ∼ = H * (A * (G/Γ)) also holds if G is completely solvable (i.e. G is solvable and for any g ∈ G the all eigenvalues of the adjoint operator Ad g are real). Thus we can extend this theorem for completely solvmanifolds. However for a general solvmanifold G/Γ, the isomorphism H * ( coE g * ) ∼ = H * (A * coE (G/Γ)) does not hold and we can't compute the coeffective cohomology by using of g * .
In [2] Baues constructed compact aspherical manifolds M Γ such that the class of these aspherical manifolds contains the class of solvmanifolds and showed that the de Rham cohomology of these aspherical manifolds can be computed by certain finite dimensional cochain complexes. In next section by using of Baues's results, we will show a generalization of Theorem 3.1.

4.
Main results: Coeffective cohomology of aspherical manifolds with torsion-free virtually polycyclic fundamental groups 4.1. Notation and conventions. Let k be a subfield of C. A group G is called a k-algebraic group if G is a Zariski-closed subgroup of GL n (C) which is defined by polynomials with coefficients in k.
Let G(k) denote the set of k-points of G and U(G) the maximal Zariski-closed unipotent normal k-subgroup of G called the unipotent radical of G. If G consists of semi-simple elements, we call G a d-group. Let U n (k) denote the n × n k-valued upper triangular unipotent matrix group.

Baues's results. A group Γ is called polycyclic if it admits a sequence
We define an infra-solvmanifold as a manifold of the form G/∆ where G is a simply connected solvable Lie group, and ∆ is a torsion free subgroup of Aut(G) ⋉ G such that for the projection p : Aut(G) ⋉ G → Aut(G) p(∆) is contained in a compact subgroup of Aut(G). By a result of Mostow in [12], the fundamental group of an infra-solvmanifold is virtually polycyclic(i.e it contains a finite index polycyclic subgroup). In particular, a lattice Γ of a simply connected solvable Lie group G is a polycyclic group with rank Γ = dim G(see [15]).
Let k be a subfield of C. Let Γ be a torsion-free virtually polycyclic group. For a finite index polycyclic subgroup ∆ ⊂ Γ, we denote rank Γ = rank ∆.
Definition 4.1. We call a k-algebraic group H Γ a k-algebraic hull of Γ if there exists an injective group homomorphism ψ : Γ → H Γ (k) and H Γ satisfies the following conditions: Let Γ be a torsion-free virtually polycyclic group and H Γ the Q-algebraic hull of Γ. Denote H Γ = H Γ (R). Let U Γ be the unipotent radical of H Γ and T a maximal d-subgroup. Then H Γ decomposes as a semi-direct product H Γ = T ⋉ U Γ see cite[Proposition 2.1]B. Let u be the Lie algebra of U Γ . Since the exponential map exp : u −→ U Γ is a diffeomorphism, U Γ is diffeomorphic to R n such that n = rank Γ. For the semi-direct product H Γ = T ⋉ U Γ , we denote φ : T → Aut(U Γ ) the action of T on U Γ . Then we have the homomorphism α : By the property (2) in Definition 4.1, φ is injective and hence α is injective.
Lemma 4.5. For p ≥ n − 1, the linear map ω∧ : ( Proof. First we notice that the map ω∧ : . Since T is d-group, for t ∈ T the t-action on u * is diagonalizable (see [2]). Hence we have a decomposition p u * = A p ⊕ B p such that A p is the subspace of t-invariant elements and B p is its complement. Since the t-action is diagonalizable, we have a basis {x 1 , . . . x 2n } of u * ⊗ C such that the t-action is represented by a diagonal matrix. Then we have If a kl = 0, we have Thus ω ∧ x i1 ∧ · · · ∧ x ip ∈ B p+2 ⊗ C. By this we have (ω ∧ B p ) ⊂ B p+2 . Since T acts semi-simply on p u * , we consider the decomposition p u * = ( p u * ) T ⊕ C p such that C p is a complement of ( p u * ) T for T -action. By the above argument we have (ω ∧ C p ) ⊂ C p+2 . Clearly we have (ω ∧ ( p u * ) T ) ⊂ ( p+2 u * ) T . Since for p ≥ n − 1 the map ω∧ : Thus we have ω ∧ ( p u * ) T = ( p+2 u * ) T . Hence the lemma follows.
By this lemma and Proposition 2.2, we have: Theorem 4.6. Let Γ be a torsion-free virtually polycyclic group and M Γ the standard Γ-manifold with a symplectic form ω such that ω ∈ ( u * ) T . Then for p ≥ n, the inclusion Φ :   Proof. If U Γ is abelian, then the differential of u * is 0. Hence we have

This gives
Hence by the above theorem the corollary follows.
In [9] the author showed the following theorem.
(2) Γ is a finite extension group of a lattice of a Lie group G = R n ⋉ φ R m such that the action φ : R n → Aut(R m ) is semi-simple.
Hence we have: Under the same assumption of Theorem 4.6, if Γ satisfies the condition (2) in Theorem 4.8, then for p ≥ n we have an isomorphism ). Remark 2. In fact by Arapura and Nori's theorem([1]) a virtually polycyclic group Γ must be virtually abelian if the standard Γ-manifold is Kähler. Therefore G/Γ is finitely covered by a torus and the assumptions of 4.8 are satisfied. By Arapura and Nori's theorem, if a solvmanifold G/Γ admits a Kähler structure, then G is (I)-type (i.e. for any g ∈ G all eigenvalues of the adjoint operator Ad g have absolute value 1). Thus in the above corollary if G is not (I)-type, then M Γ does not admit a Kähler structure. The author gave such non-Kähler examples in [9].

examples
Example 1. First we give examples of solvmanifolds such that H p (A * coE (M Γ )) ∼ =H p (A * (M Γ )) by using of Corollary 4.9. We notice that if a solvmanifold G/Γ has a symplectic form ω then we have a closed two form ω 0 ∈ ( u * ) T which is homologous to ω and ω 0 is also a symplectic form as we Then it is known that G has a left-invariant symplectic form and a lattice Γ (see [13]). Thus we have a symplectic form ω ∈ ( u * ) T and by Corollary 4.9 we have an isomorphism H p (A * coE (G/Γ)) ∼ =H p (A * (G/Γ)). Remark 3. G is not completely solvable. In fact the de Rham cohomology of G/Γ varies according to a choice of a lattice Γ. Thus it is not easy to compute the coeffective cohomology of G/Γ by using of g * .
Remark 4. G is not (I)-type and hence G/Γ does not admit a Kähler structure.
Example 2. We give an example of a symplectic manifold M Γ such that the isomorphism Then we have H Γ = {±1} ⋉ U 3 (R) such that [9,Section 7]). The dual space of the Lie algebra u of U Γ is given by u * = x 1 , x 2 , x 3 such that the differential is given by Then we have ( u * ) {±1} = x 1 , x 2 ∧ x 3 . By this the differential on ( u * ) {±1} is 0. We consider the product M Γ × M Γ for this Γ. Then by the cochain complex ( u * ) {±1} ⊗ ( u * ) {±1} = x 1 , x 2 ∧ x 3 ⊗ y 1 , y 2 ∧y 3 we can compute the de Rham cohomology and coeffective cohomology of M Γ ×M Γ where we denote y 1 , y 2 , y 3 the copy of x 1 , x 2 , x 3 . We have a symplectic form ω = x 1 ∧ y 1 + x 2 ∧ x 3 + y 2 ∧ y 3 on M Γ × M Γ . Then we have: Proposition 5.1. For p ≥ n we have an isomorphism ). Proof. Since the differential on ( u * ) {±1} ⊗ ( u * ) {±1} is not 0 as above, the proposition follows as the proof of Corollary 4.7.
Remark 5. M Γ is finitely covered by a quotient of U 3 (R) by a lattice. Thus M Γ × M Γ is finitely covered by the product of such nilmanifolds. The de Rham cohomology and coeffective cohomology of this covering space are computed by u * ⊗ u * . This space does not satisfy the isomorphism as this proposition. Indeed x 1 ∧ x 2 ∧ y 2 ∧ y 3 is coeffective and its coeffective cohomology class is not 0. But we have d(x 3 ∧ y 2 ∧ y 3 ) = x 1 ∧ x 2 ∧ y 2 ∧ y 3 and hence its de Rham cohomology class is 0. Thus we have