Primitive decomposition of Bott–Chern and Dolbeault harmonic ( k , k ) -forms on compact almost Kähler manifolds

We consider the primitive decomposition of ∂, ∂ , Bott–Chern and Aeppli-harmonic ( k , k ) -forms on compact almost Kähler manifolds ( M , J , ω) . For any D ∈ { ∂, ∂, BC , A } , it is known that the L k P 0 , 0 component of ψ ∈ H k , kD is a constant multiple of ω k up to real dimension 6. In this paper we generalise this result to every dimension. We also deduce information on the components L k − 1 P 1 , 1 and L k − 2 P 2 , 2 of the primitive decomposition. Focusing on dimension 8, we give a full description of the spaces H 2 , 2BC and H 2 , 2A , from which follows H 2 , 2BC ⊆ H 2 , 2 ∂ and H 2 , 2A ⊆ H 2 , 2 ∂ . We also provide an almost Kähler 8-dimensional example where the previous inclusions are strict and the primitive components of a harmonic form ψ ∈ H k , kD are not D - harmonic, showing that the primitive decomposition of ( k , k ) -forms in general does not descend to harmonic forms.


Introduction
A recent answer to a question of Kodaira and Spencer, [3,Problem 20], shows that the dimension of the space of Dolbeault harmonic forms depends on the choice of the metric on a given compact almost complex manifold, see [5,6].
The primitive decomposition of harmonic forms has proven to be useful in describing the spaces of harmonic (1, 1)-forms in dimension 4. In the case of Dolbeault harmonic forms it has been used to show that h 1,1 ∂ is either equal to b − or b − + 1, depending on the choice of metric, see [4,5,12].Similarly, for Bott-Chern harmonic forms, it yields h 1,1  BC ∶= dim C H 1,1 BC = b − + 1 for all metrics, see [4,10].See [8,11,13] for other related results and [7,15] for two surveys on the subject.
In this paper, we explore what the primitive decomposition can tell us about harmonic (k, k)-forms in higher dimensions.We start by considering a 2n-dimensional almost Hermitian manifold (M, J, ω).The almost complex structure J induces the bidegree decomposition on the space of complex valued k-forms Additionally, the almost Hermitian structure induces the primitive decomposition on the space of k-forms given by where L ∶= ω∧, Λ ∶= * −1 L * and P s ∶= ker Λ ∩ A s is the space of primitive s-forms, for s ≤ n (see e.g., [14,p. 26,Théorème 3]).These two decompositions are compatible with each other.
In fact, for Kähler manifolds, i.e., when J is integrable and dω = 0, the primitive decomposition passes to the space of d-harmonic (p, q)-forms, denoted by H p,q d (M, J) ∶= ker ∆ d ∩ A p,q , namely (1) H p,q d = ⊕ r≥max(p+q−n,0) L r (H p−r,q−r d ∩ P p−r,q−r ).
where P p,q ∶= P m C ∩ A p,q .On Kähler manifolds, we also know that H p,q d = H p,q D for all D ∈ {∂, ∂, BC, A} (see Section 2 for the definitions of these spaces), therefore we have (2) H p,q D = ⊕ r≥max(p+q−n,0) L r (H p−r,q−r D ∩ P p−r,q−r ).
We remark that (1) and ( 2) have a cohomological meaning in the Kähler setting.
These two results are sufficient to prove that either (2) or its dual through the Hodge * operator hold for any space of D-harmonic (k, k)-forms on any compact almost Kähler manifold with dimension up to 6.This raises the following question, which we shall answer in this paper: does (2) (or its * dual) hold for (k, k)-forms in general for compact almost Kähler manifolds with dimension 8 or greater?We note that (2) has been shown to fail for dimension 6 in bidegree (2, 1) in [1, Proposition 5.1] for D ∈ {∂, ∂} and in [9, Proposition 5.1] for D ∈ {BC, A}.
We also remark that the almost Kähler assumption is necessary for this kind of primitive harmonic decomposition.To see that this is the case in dimension 4 we refer the reader to [12,10].
The structure of this paper is as follows.In Section 2 we give a brief overview of some of the basic results which will be used throughout the paper.In Section 3 we show that Theorem 1.2 may be partially extended to (k, k)-forms.Theorem 3.8.Let (M, J, ω) be a compact almost Kähler manifold of real dimension 2n.For any k ∈ N we have We also consider the 8-dimensional case in more detail, yielding the following description.
Corollary 3.10.Let (M, J, ω) be a compact almost Kähler manifold of real dimension 8.We have In Section 4 we show that Theorem 1.1 may also be partially extended to (k, k)forms.
Theorem 4.3.Let (M, J, ω) be a compact almost Kähler manifold of real dimension 2n.For any k ∈ N we have Looking at the special case of this corollary in dimension 8, along with Corollary 3.10, we are able to deduce the following Corollary 4.6.Let (M, J, ω) be a compact almost Kähler manifold of real dimension 8.We have Finally in Section 5, we consider a non left invariant almost Kähler structure on the 8-dimensional torus T 8 = Z 8 R 8 .We use this example to show that there exists a (2, 2)-form contained in H 2,2 ∂ but not in H 2,2 BC , and likewise there exists a A .We also show that there exists a form

BC
whose components with respect to the primitive decomposition, α ∈ P We also consider another 8-dimensional compact nilmanifold, focusing on the subspace of left invariant harmonic forms in H 2,2 D .We show that these spaces satisfy (2) for all D ∈ {∂, ∂, BC, A}.Furthermore, we show that in this example these spaces are in fact all equal and have dimension 16.

Preliminaries
Throughout this paper, we will only consider connected manifolds without boundary.Let (M, J) be an almost complex manifold of dimension 2n, i.e., a 2n-differentiable manifold endowed with an almost complex structure J, that is J ∈ End(T M ) and J 2 = − id.The complexified tangent bundle T C M = T M ⊗ C decomposes into the two eigenspaces of J associated to the eigenvalues i, −i, which we denote respectively by T 1,0 M and T 0,1 M , giving us Denoting by Λ 1,0 M and Λ 0,1 M the dual vector bundles of T 1,0 M and T 0,1 M , respectively, we set to be the vector bundle of (p, q)-forms, and let A p,q = Γ(M, Λ p,q M ) be the space of smooth sections of Λ p,q M .We denote by ) be a smooth function on M with complex values.Its differential df is contained in A 1 ⊗ C = A 1,0 ⊕ A 0,1 .On complex 1-forms, the exterior derivative acts as Therefore, it turns out that the derivative operates on (p, q)-forms as d ∶ A p,q → A p+2,q−1 ⊕ A p+1,q ⊕ A p,q+1 ⊕ A p−1,q+2 , where we denote the four components of d by From the relation d 2 = 0, we derive We also define the operator If the almost complex structure J is induced from a complex manifold structure on M , then J is called integrable.Recall that J is integrable if and only if the exterior derivative decomposes into d = ∂ + ∂.
A Riemannian metric g on M which is preserved by J, i.e. g(J⋅, J⋅) = g(⋅, ⋅), is called almost Hermitian.Let g be an almost Hermitian metric, the 2-form ω such that is called the fundamental form of g.We will call (M, J, ω) an almost Hermitian manifold.We denote by h the Hermitian extension of g on the complexified tangent bundle T C M , and by the same symbol g the C-bilinear symmetric extension of g on T C M .Also denote by the same symbol ω the C-bilinear extension of the fundamental form ω of g on T C M .Thanks to the elementary properties of the two extensions h and g, we may want to consider h as a Hermitian operator Let (M, J, ω) be an almost Hermitian manifold of real dimension 2n.Denote the extension of h to (p, q)-forms by the Hermitian inner product ⟨⋅, ⋅⟩.Let * ∶ A p,q → A n−q,n−p be the C-linear extension of the standard Hodge * operator on Riemannian manifolds with respect to the volume form Vol = ω n n! , i.e., * is defined by the relation α ∧ * β = ⟨α, β⟩ Vol ∀α, β ∈ A p,q .Integrating the pointwise Hermitian inner product on the manifold, we get the standard L 2 product here denoted by ⟪α, β⟫ which is surely well defined if M is compact.Then the operators is the Hodge Laplacian, and, as in the integrable case, set If M is compact, then we easily deduce the following relations which characterize the spaces of harmonic forms A , defined as the spaces of forms which are in the kernel of the associated Laplacians.All these Laplacians are elliptic operators on the almost Hermitian manifold (M, J, ω) (cf.[3], [10]), implying that all the spaces of harmonic forms are finite dimensional when the manifold is compact.Now we introduce some notation and recall some well known facts about primitive forms.We denote by Then we have the following vector bundle decomposition (see e.g., [14, p. 26, Théorème 3]) where we use to denote the bundle of primitive s-forms.For any given β ∈ P k M , we have the following formula (cf.[14, p. 23, Théorème 2]) involving the Hodge * operator and the Lefschetz operator We recall that the map Furthermore, the decomposition above is compatible with the bidegree decomposition on the bundle of complex k-forms Λ k C M induced by J, that is where P p,q M = P k C M ∩ Λ p,q M. In fact, we have (7) Λ p,q M = ⊕ r≥max(p+q−n,0) L r (P p−r,q−r M ).
Finally, let us set P s ∶= Γ(M, P s M ) and P p,q ∶= Γ(M, P p,q M ).

Primitive decomposition of Bott-Chern harmonic (k, k)-forms
In order to prove our main result, we will need the following lemmas.The next one is well known, see for instance [9,Theorem 3.2] or [10,Theorem 4.3].We include an outline of the proof here for the convenience of the reader.
The wedge product of ω n−1 with ϕ ij is zero unless i = j, therefore where R is a differential operator involving at most first order derivatives.Setting this to zero tells us that f is in the kernel of a strongly elliptic differential operator.
In conjunction with the compactness of M , this implies that f must be constant by the maximum principle.
By (6), we have the formula We want to compare (9) and (10).Note that therefore we can compute the wedge product between (9) and ω to the power n If we take the wedge product of ω k−1 with both equations ( 10) and ( 11), we find In the same way, by (8).Now, thanks to the last two equations, we easily deduce ω n−1 ∧ ∂∂α 0,0 = 0 and ω n−2 ∧ ∂∂α 1,1 = 0.In particular, this yields that α 0,0 ∈ C is a complex constant by Lemma 3.1.
If we take the wedge product of ω k−2 with both equations ( 10) and ( 11), we find by ( 8).In the same way, by (8).Now, thanks to the last two equations, we easily deduce and ω n−4 ∧ ∂∂α 2,2 = 0, which is the claim.Remark 3.3.If we take the wedge product of ω k−l , for 3 ≤ l ≤ k − 1, with both equations (10) and (11), we find similar sums, but this time we have three or more addends.This does not imply, in general, that every addend is equal to 0.
Assume now that ∂ψ = 0 and α 0,0 ∈ C. Since dω = 0, it follows If we take the wedge product of ω n−k−1 and ( 13), we find by (8), and this ends the proof.Remark 3.5.If we take the wedge product of ω n−k−l , for 2 ≤ l ≤ n − k − 1, with both equations ( 12) and (13), we find similar sums, but this time we have two or more addends.This does not imply, in general, that every addend is equal to 0.
Finally, in the following lemma we study the first order differential conditions in the characterisation (4) of Aeppli harmonic forms.Lemma 3.6.Let (M, J, ω) be a compact almost Kähler manifold of real dimension 2n.For any k ∈ N such that 2k ≤ n, write any (k, k)-form ψ as where every (m, m)-form α m,m is primitive.Assume that α 0,0 ∈ C is a complex constant.If ∂ * ψ = 0, then ∂α 1,1 is primitive, i.e., Assume that ∂ * ψ = 0 and α 0,0 ∈ C. Since dω = 0, it follows If we take the wedge product of ω k−1 and ( 14), we find by (8), and this is equivalent to the first claim.Now, assume that ∂ * ψ = 0 and α 0,0 ∈ C. Since dω = 0, it follows If we take the wedge product of ω k−1 and (15), we find by (8), and this is equivalent to the second claim.
Since the coefficient of ω k is constant in the primitive decomposition of any Bott-Chern or Aeppli harmonic (k, k)-form, we can state the following characterisations of the spaces of Bott-Chern and Aeppli harmonic (k, k)-forms.Theorem 3.8.Let (M, J, ω) be a compact almost Kähler manifold of real dimension 2n.For any k ∈ N we have Let us consider the primitive decomposition of a Bott-Chern or Aeppli harmonic (k, k)-form ψ, i.e., Thanks to Theorem 3.7, we know that the coefficient of ω k , denoted by α 0,0 , is a complex constant.Since ω n−2m+1 ∧ α m,m = 0, therefore for any m ≥ 1.This proves the two inclusions ⊆.The other two inclusions ⊇ are trivial.
Remark 3.9.When we consider the case of Theorem 3.8 when k = 1, we recover the results of Tardini and the second author, [9, Theorems 3.2 and 3.3].Namely, we have a complete decomposition into primitive harmonic forms A ∩ P 1,1 .Corollary 5.2 will show that a complete decomposition into primitive harmonic forms does not hold in higher bidegrees (k, k) with k ≥ 2.
The previous corollary can be further specialized in real dimension 8 for bidegree (2, 2).Corollary 3.10.Let (M, J, ω) be a compact almost Kähler manifold of real dimension 8.We have Let us prove the Bott-Chern case.The Aeppli case is proved by a similar argument.By Theorem 3.8, we know Since L∂α = −∂β and they are L 2 orthogonal, they must both be equal to zero.Now, by the Lefschetz isomorphism, L∂α = 0 if and only if ∂α = 0.This ends the proof.

Primitive decomposition of Dolbeault Harmonic (k, k)-forms
The next theorem yields similar conclusions for the Dolbeault case to the ones in the Bott-Chern and Aeppli case.Theorem 4.1.Let (M, J, ω) be a compact almost Kähler manifold of real dimension 2n.Let ψ denote a (k, k)-form, for some k ∈ N. We can write with α m,m ∈ P m,m .If ψ is Dolbeault harmonic then α 0,0 ∈ C is a complex constant and ∂α 1,1 , ∂α 1,1 are primitive.
Proof.We start by considering the case when 2k ≤ n, using a similar argument to the one used in [1,Theorem 3.4].
Note that ψ is Dolbeault harmonic if and only it satisfies ∂ψ = 0 and ∂ * ψ = 0. Since ω is almost Kähler, when we write these conditions out using the primitive decomposition of ψ we get Then, by taking the wedge product of ω n−k−1 with equation ( 16) we find that since ω n−2m+1 ∧ α m,m = 0 for all m ∈ N. Similarly, by taking the wedge product of ω k−1 with equation (17) we find Multiplying this by We can then sum the equations ( 18) and ( 19), to get (20) Now, by making use of the operator and the fact that µ = µ = 0 when acting on an (n − 1, n − 1)-form, we can rewrite equation (20) as Applying −id c the right hand side vanishes and we are left with In particular, we have ω n−1 ∧ ∂∂α 0,0 = which implies that α 0,0 is constant by Lemma 3.1.Now, looking at (20), we deduce that ∂α 1,1 and ∂α 1,1 are primitive.
The result in the case when 2k ≥ n follows simply from the first case by Serre duality.Namely, we have for some ψ ∈ H n−k,n−k ∂ , and since 2(n − k) ≤ n we conclude that α 0,0 is constant and ∂α 1,1 and ∂α 1,1 are primitive.
Remark 4.2.Note that the same statement of Theorem 4.1 also holds for ∆ ∂harmonic (k, k)-forms.The proof of this is equivalent to the one above up to conjugation.
The above result allows the primitive decomposition of (k, k)-forms to descend partially to Dolbeault harmonic (k, k)-forms in the following way, using the same proof as Theorem 3.8.Theorem 4.3.Let (M, J, ω) be a compact almost Kähler manifold of real dimension 2n.For any k ∈ N we have Namely, we have a complete decomposition into primitive harmonic forms . Corollary 5.2 will show that a complete decomposition into primitive harmonic forms does not hold in higher bidegrees (k, k) with k ≥ 2.
If we now restrict to real dimension 8, we obtain the following.Corollary 4.5.Let (M, J, ω) be a compact almost Kähler manifold of real dimension 8.We have The result follows immediately from Theorem 4.3, from the characterisation (4) of Dolbeault and ∂-harmonic forms and from formula (6).
From Corollaries 3.10 and 4.5, we deduce the following inclusions of the spaces of harmonic forms in dimension 8.

Examples
In this section we present two 8-dimensional examples of nilmanifolds and study their harmonic (2, 2)-forms.
We consider the volume form We want to show that the inclusion H BC , proving our claim.We note that the same holds for the form ϕ 1323 .By duality (see (4)), also note that * A .We also want to show that in general the primitive decomposition of (2, 2)-forms does not descend to the spaces of Bott-Chern, Aeppli, Dolbeault and ∂-harmonic forms.Namely, we want to find a (2, 2)-form where α ∈ P 1,1 and β ∈ P 2,2 , such that α and β are not Bott-Chern and ∂-harmonic.Considering ψ, the same can then be shown for the cases of Aeppli and Dolbeault.Let us consider the form ψ = 2ϕ 2144 .Its primitive decomposition is where Set β = ϕ 2144 − ϕ 2133 and α = −iϕ 21 .Then we have ∂ .Note that this also shows that the two results of Corollary 3.10 cannot be strengthened by asking that ω ∧ ∂α = ∂β = 0, instead of ω ∧ ∂α + ∂β = 0 or ω ∧ ∂α − ∂β = 0.
Summing up the results from the above example, we state the following corollary.
Corollary 5.2.There exists a compact almost Kähler manifold (M, J, ω) of real dimension 8 such that We remark that the almost Kähler structure of Example 5.1 is not left invariant with respect to the usual Lie group structure of the torus.In fact, we do not have any example of an 8-dimensional manifold with a left invariant almost Kähler structure which satisfies the conditions of Corollary 5.2.Below we present one such example of a manifold (described in [1], Example 4.3), with a left invariant almost Kähler structure.We show that in this example all left invariant harmonic forms ψ ∈ H 2,2 D have a primitive decomposition such that each component is also contained in H 2,2 D , where D ∈ {BC, A, ∂, ∂}.Example 5.3.We start by defining Then, if we let Γ ⊂ H(2, 1) be the subgroup of elements with integer valued entries, we can define the compact 8-manifold X ∶= Γ H(2, 1) × T 3 .A left invariant coframe on X can be given by An almost Hermitian structure is then defined so that A , we find that these spaces are all equal to the space of left invariant forms in H 2,2 ∂ .We claim that this implies that the primitive decomposition descends to harmonic (2, 2)-forms in all of these spaces, i.e., if ψ ∈ H 2,2 D is left invariant then ψ ∈ C ω 2 ⊕ L P To see why this is the case, consider the (2, 2)-form ψ ∈ A 2,2 (X) with primitive decomposition ψ = cω 2 + α ∧ ω + β, where c ∈ C, α ∈ P 1,1 , β ∈ P 2,2 .Let ψ be contained in both H