On Cohomological Decomposability of Almost-K\"ahler Structures

We study the J-invariant and J-anti-invariant cohomological subgroups of the de Rham cohomology of a compact manifold M endowed with an almost-K\"ahler structure (J, \omega, g). In particular, almost-K\"ahler manifolds satisfying a Lefschetz type property, and solvmanifolds endowed with left-invariant almost-complex structures are investigated.


Introduction
Cohomological properties of compact complex, and, more in general, almostcomplex, manifolds have been recently studied by many authors, see, e.g., [3], respectively [11,12], and the references therein. The study of the cohomology of almost-complex manifolds is motivated, in particular, by a question of Donaldson's, [10,Question 2], relating the tamed and compatible symplectic cones of a compact 4-dimensional almost-complex manifold, see, e.g., [20], and by the analogous question arising for compact higher dimensional complex manifolds, see [20, page 678] and [26,Question 1.7]. (We recall that a symplectic structure ω on a manifold M is said to tame an almost-complex structure J if ω x (u x , J x u x ) > 0 for any x ∈ M and for any u ∈ T x M \ {0}, and it is said compatible with J if g := ω(·, J · ·) is a J-Hermitian metric; in the latter case, the triple (J, ω, g) is called an almost-Kähler structure on M .) Following T.-J. Li and the third author, [20], an almost-complex structure J on a 2n-dimensional manifold M is called C ∞ -pure-and-full if H 2 dR (M ; R) = H C 3 , * be the real manifold underlying the Iwasawa manifold. Then there exists an almost-Kähler structure (J, ω, g) on X which is C ∞ -pure and non-C ∞ -full. Furthermore, the Lefschetz type operator L ω := ω ∧ · : ∧ 2 M → ∧ 4 M of the almost-Kähler structure (J, ω, g) does not take gharmonic 2-forms to g-harmonic 4-forms.
In studying cohomological decomposition of the de Rham cohomology of almost-Kähler manifolds, the third author introduced a Lefschetz type property for 2-forms, see Definition 2.2. Such a property is stronger than the Hard Lefschetz Condition on 2-classes, namely, the property that [ω] n−2 ⌣ · : H 2 dR (M ; R) → H 2n−2 dR (M ; R) is an isomorphism, where 2n := dim M .
We study such a Lefschetz type property on almost-Kähler manifolds (M, J, ω, g) in relation to the existence of a cohomological decomposition of H 2 dR (M ; R). More precisely, we prove the following result. Theorem 2.3. Let (M, J, ω, g) be a compact almost-Kähler manifold. Suppose that there exists a basis of H 2 dR (X; R) represented by g-harmonic 2-forms which are of pure type with respect to J. Then the Lefschetz type property on 2-forms is satisfied.
Note that, by the hypothesis, it follows,in particular, that J is C ∞ -pure-and-full and pure-and-full, [15,Theorem 3.7]. Note also that A. Fino and the second author provided in [15] several examples of compact non-Kähler solvmanifolds admitting a basis of harmonic representatives of pure-type with respect to the almost-complex structure. In [13, §2], T. Drǎghici, T.-J. Li, and the third author ask whether such a Lefschetz type property on 2-forms is actually equivalent to C ∞ -fullness for every almost-Kähler nilmanifold and solvmanifold, without any further assumption; Theorem 2.
Then the space ∧ r M of real smooth differential r-forms decomposes as C ∞ -pure-and-full and pure-and-full almost-complex structures. As in [20], we set the following definition. ). An almostcomplex structure J on a manifold M is said to be According to the previous notation, we will write Similar definitions in terms of currents can be given, introducing the notion of pure-and-full almost-complex structure: we refer to [20, §2.2.2] for further details and results. More precisely, on an almost complex manifold (M, J), the space Let Z J (2,0),(0,2) and Z J (1,1) denote the spaces of real d-closed currents of bidimension (2, 0) + (0, 2), respectively (1, 1), and B J (2,0),(0,2) and B J (1,1) denote the spaces of real d-exact currents of bi-dimension (2, 0) + (0, 2), respectively (1, 1). Denote by B the space of boundaries. Let, as in [20], We recall the following definition. (M ) R of the de Rham cohomology can be viewed as an analogue of the Dolbeault cohomology groups for non-integrable almost-complex structures.
In [11,Theorem 2.3] it is proven the following result. This is no more true in dimension higher than 4: in [15, Example 3.3], a compact non-C ∞ -pure almost-complex structure on a 6-dimensional nilmanifold is constructed. Therefore, the previous theorem can be considered a sort of Hodge decomposition theorem in the non-Kähler case.

Cohomological properties of almost-Kähler manifolds
2.1. Lefschetz type property on almost-Kähler manifolds with pure-type harmonic representatives. Given a compact 2n-dimensional almost-Kähler manifold (M, J, ω, g), we are interested in studying the property of being C ∞ -pure-andfull.
Firstly we recall the following result. . If J is an almostcomplex structure on a compact manifold M and J admits a compatible symplectic structure, then J is C ∞ -pure.
Furthermore, A. Fino and the second author proved that an almost-Kähler manifold admitting a basis of harmonic 2-forms whose elements are of pure type with respect to the almost-complex structure is C ∞ -pure-and-full and pure-and-full, [15,Theorem 3.7]; they also provided several examples of compact non-Kähler solvmanifolds satisfying the above assumption in [15].
To the purpose of studying the property of being C ∞ -pure-and-full on almost-Kähler manifolds, we recall the following definition.
Furthermore, we recall some notions and results from [6,22,27], see also [23,7]. Let (M, ω) be a compact 2n-dimensional symplectic manifold. Extend ω −1 : T * M → T M to the whole exterior algebra of T * M . For any k ∈ N, the symplectic ⋆ ω operator is defined as One can prove that ⋆ 2 ω = id ∧ • M , [6, Lemma 2.1.2]. For any k ∈ N, define the symplectic co-differential operator this operator has been studied by J.-L. Brylinski in [6] for Poisson manifolds; in the context of generalized complex geometry, see, e.g., [16], it can be interpreted as the symplectic counterpart of the operator d c := − i ∂ − ∂ in complex geometry, see [7]. By definition, (M, ω) satisfies the Hard Lefschetz Condition if, for each k ∈ N, the map [22,Corollary 2], and, independently, D. Yan, [27, Theorem 0.1], proved that, given a compact symplectic manifold (M, ω), any de Rham cohomology class has a (possibly non-unique) ω-symplectically harmonic representative (that is, a d-closed δ ω -closed representative) if and only if the Hard Lefschetz Condition holds.
We can now prove the following result. Proof. Recall that, on a 2n-dimensional almost-Kähler manifold (M, J, ω, g), the Hodge * g operator and the symplectic ⋆ ω operator are related by ⋆ ω = * g J, [6, Theorem 2.4.1, Remark 2.4.4]. Therefore, for forms of pure type with respect to J, the properties of being g-harmonic and of being ω-symplectically harmonic are equivalent. The theorem follows noting that, [27,Lemma 1.2], [L ω , d] = 0 and [L ω , δ ω ] = d, hence L ω sends ω-symplectically harmonic 2-forms (of pure type with respect to J) to ω-symplectically harmonic (2n−2)-forms (of pure type with respect to J).
Remark 2.4. We note that if (M, J, ω, g) is a compact 2n-dimensional almost-Kähler manifold satisfying the Lefschetz type property on 2-forms and J is C ∞ -full, then J is C ∞ -pure-and-full and pure-and-full.
Indeed, we have already remarked that J is C ∞ -pure, see Proposition 2.1. Moreover, since J is C ∞ -full, J is also pure by [20,Proposition 2.5]. We recall now the argument in [15] to prove that J is also full.
Firstly, note that if the Lefschetz type property on 2-forms holds, then ω n−2 ⌣ · : • setting g α (·, ·) := ω α (·, J α ·), the triple (J α , ω α , g α ) gives rise to a family of left-invariant almost-Kähler structures on N ; • denoting by E 4 α := e 4 , E 5 α := e 5 , E 6 α := e 6 , then E 1 α , . . . , E 6 α is a g α -orthonormal co-frame on N ; with respect to this new co-frame, we easily obtain the following structure equations: Then, where we have listed the g α -harmonic representatives instead of their classes). Note that the listed g α -harmonic representatives of H 2 dR (N ; R) are of pure type with respect to J α : hence, the almost-complex structure J α is C ∞ -pure-and-full by [15,Theorem 3.7]; in particular, note that and, by a similar computation, d * gα L ωα e 25 + e 36 = 0. This proves explicitly that ω α satisfies the Lefschetz type property on 2-forms. The nilmanifold N is not formal by a theorem of K. Hasegawa's, [17,Theorem 1,Corollary]. The non-formality of M can be also proved by giving a non-zero triple Massey product on N , see [9]: since we get that the triple Massey product does not vanish, and hence N is not formal.
In summary, we have proven the following result.

3.
Almost-Kähler C ∞ -pure-and-full structures 3.1. The Nakamura manifold of completely solvable type. Take A ∈ SL(2; Z) with two different real eigenvalues e λ and e −λ with λ > 0, and fix P ∈ GL(2; R) such that P AP −1 = diag e λ , e −λ . For example, take where T 2 C is the 2-dimensional complex torus T 2 C := C 2 P Z[i] 2 and T 1 acts on R × T 2 C as T 1 x 1 , x 3 , x 4 , x 5 , x 6 := x 1 + λ, e −λ x 3 , e λ x 4 , e −λ x 5 , e λ x 6 . The manifold M 6 can be seen as a compact quotient of a completely-solvable Lie group by a discrete co-compact subgroup, [14, Example 3.1]; (denote the Lie algebra naturally associated to the completely-solvable Lie group of M 6 by g). Using coordinates x 2 on S 1 , x 1 on R and x 3 , x 4 , x 5 , x 6 on T 2 C , we set It is straightforward to check that J is integrable. Being M 6 a compact quotient of a completely-solvable Lie group, one computes the ; C ≃ C ϕ 123 , ϕ 132 , ϕ 123 , ϕ 123 , ϕ 213 , ϕ 312 , ϕ 231 , ϕ123 (for the sake of clearness, we write, for example, ϕ AB in place of ϕ A ∧φ B and we list the harmonic representatives with respect to the metric g := 3 j=1 ϕ j ⊙φ j instead of their classes). Therefore, M 6 is geometrically formal, i.e., the product of g-harmonic forms is still g-harmonic, and therefore it is formal, namely the de Rham complex of M is formal as a differential graded algebra, see, e.g., [9]. Furthermore, it can be easily checked that ω := e 12 + e 34 + e 56 gives rise to a symplectic structure on M 6 satisfying the Hard Lefschetz Condition. We obtain the following result. Note also thatω := i 2 ϕ 11 + ϕ 22 + ϕ 33 is not d-closed but dω 2 = 0, from which it follows that the manifold M 6 admits a balanced metric.
Moreover, since M 6 is a compact quotient of a completely-solvable Lie group, by the K. Hasegawa's theorem [18, Main Theorem], we have the following result, see also [14,Theorem 3.3]. (We recall that a compact complex manifold is said to belong to class C of Fujiki if it admits a proper modification from a Kähler manifold.)

3.
3. An almost-Kähler structure on the Nakamura manifold. By K. Hasegawa's theorem [18,Main Theorem], any integrable complex structure on M 6 (for example, the J defined in §3.2) does not admit any symplectic structure compatible with it. Therefore, we consider the almost-complex structure J ′ defined by as a co-frame for the space of (1, 0)-forms on M 6 , J ′ , one can compute from which it is clear that J ′ is not integrable. Note that the J ′ -compatible 2-form ω ′ := e 12 + e 34 + e 56 is d-closed. Hence, M 6 , J ′ , ω ′ is an almost-Kähler manifold. Moreover, recall that where we have listed the harmonic representatives with respect to the metric g ′ := 6 j=1 e j ⊙e j instead of their classes; note that the listed g ′ -harmonic representatives are of pure type with respect to J ′ . Therefore, J ′ is obviously C ∞ -full; it is also C ∞pure by [15,Proposition 3.2] or [11,Proposition 2.8], see Proposition 2.1. Moreover, since any cohomology class in H + [15,Theorem 3.7] we have that J ′ is also pure-and-full. One can explicitly check that the Lefschetz type operator L ω ′ : ∧ 2 M 6 → ∧ 4 M 6 introduced in §2 takes g ′ -harmonic 2-forms to g ′ -harmonic 4-forms, since L ω ′ e 12 = e 1234 + e 1256 = * g ′ e 34 + e 56 , L ω ′ e 36 = e 1236 = * g ′ e 45 , L ω ′ e 34 = e 1234 + e 3456 = * g ′ e 12 + e 56 , L ω ′ e 45 = e 1245 = * g ′ e 36 , L ω ′ e 56 = e 1256 + e 3456 = * g ′ e 12 + e 34 .
Resuming, we have shown the following result. Proposition 3.3. Let M 6 be the Nakamura manifold. Then there exist a complex structure J and an almost-Kähler structure (J ′ , ω ′ , g ′ ), both of which are C ∞ -pureand-full and pure-and-full. Furthermore, the Lefschetz type operator of the almost-Kähler structure (J ′ , ω ′ , g ′ ) takes g ′ -harmonic 2-forms to g ′ -harmonic 4-forms.
Considering the standard complex structure induced by the one on C 3 and setting ϕ 1 , ϕ 2 , ϕ 3 as a global co-frame for the (1, 0)-forms on X, by A. Hattori's theorem [19,Corollary 4.2], see Theorem 5.3, one gets that where we have listed the harmonic representatives with respect to the metric g := We easily get that . We claim that the previous inclusions are actually equalities, and in particular that J is a non-C ∞ -full almost-Kähler structure on X. Indeed, we firstly note that, by [15,Proposition 3.2] or [11, Proposition 2.8], see Proposition 2.1, J is C ∞ -pure, since it admits a symplectic structure compatible with it. Moreover, we recall that a C ∞ -full almost-complex structure is also pure by [20, Proposition 2.30] and therefore it satisfies also that (X) R ∋ e 3456 , therefore (1) does not hold, and hence J is not C ∞ -full.
Let L ω be the Lefschetz type operator of the almost-Kähler structure (J, ω, g). Then, we have L ω e 12 = e 1234 = d e 245 , i.e., L ω does not take g-harmonic 2-forms in g-harmonic 4-forms.
Hence, we have proved the following result. 3 C 3 , * be the real manifold underlying the Iwasawa manifold. Then there exists an almost-Kähler structure (J, ω, g) on X which is C ∞ -pure and non-C ∞ -full. Furthermore, the Lefschetz type operator of the almost-Kähler structure (J, ω, g) does not take g-harmonic 2-forms to g-harmonic 4-forms.

Almost-complex manifolds with large anti-invariant cohomology
Given an almost-complex structure J on a compact manifold M , it is natural to ask how large the cohomology subgroup H