Primitive decompositions of Dolbeault harmonic forms on compact almost-K\"ahler manifolds

Let $(X,J,g,\omega)$ be a compact $2n$-dimensional almost-K\"ahler manifold. We prove primitive decompositions of $\partial$-, $\overline{\partial}$-harmonic forms on $X$ in bidegree $(1,1)$ and $(n-1,n-1)$ (such bidegrees appear to be optimal). We provide examples showing that in bidegree $(1,1)$ the $\partial$- and $\overline{\partial}$-decompositions differ.


Introduction
In complex geometry, the Dolbeault cohomology plays a fundamental role in the study of complex manifolds, and a classical way to compute it on compact complex manifolds is through the use of the associated spaces of harmonic forms.More precisely, if X is a complex manifold, then the exterior derivative d splits as ∂ + ∂, and such operators satisfy ∂2 = ∂ 2 = ∂ ∂ + ∂∂ = 0. Hence, one can define the Dolbeault cohomology and its conjugate as If X is compact and we fix an Hermitian metric, then it turns out that these spaces are isomorphic to the kernel of two suitable elliptic operators, ∆ ∂ and ∆ ∂ , respectively.More precisely, denoting with H •,• ∂ (X) and H •,• ∂ (X) the spaces of harmonic forms, they have a cohomological meaning, namely , and in particular their dimensions are holomorphic invariants.
Moreover, if the Hermitian metric is Kähler, then by the Kähler identities it turns out that ∆ ∂ = ∆ ∂ and in particular A. CATTANEO, N. TARDINI, AND A. TOMASSINI therefore giving isomorphisms for the respective cohomologies, namely . The integrability assumption on the complex structure is crucial in the proof of all these results.Furthermore, a remarkable feature of Kähler geometry is that the primitive decomposition of differential forms passes to cohomology and leads to a primitive decomposition of de Rham cohomology (see, e.g., [18]).Kähler geometry is at the crossroad of complex and symplectic geometry.From the symplectic point of view we recall that Tseng and Yau [17] introduced natural cohomologies on (compact) symplectic manifolds, involving the symplectic co-differential and the exterior derivative, proving a primitive decomposition for them.
If J is a non-integrable almost-complex structure on a 2n-dimensional smooth manifold X, then the exterior derivative splits as µ + ∂ + ∂ + μ, and in particular ∂2 ̸ = 0. Hence, the standard Dolbeault cohomology and its conjugate are not well-defined.Recently, Cirici and Wilson [6] gave a definition for the Dolbeault cohomology in the non-integrable setting considering also the operator μ together with ∂.Such cohomology groups might be infinite-dimensional on compact almostcomplex manifolds as shown in [7].
On the other hand, fixing an almost-Hermitian metric g on (X, J) one can develop a Hodge theory for harmonic forms on (X, J, g) without a cohomological counterpart.More precisely, setting, similarly to the integrable case, it turns out that they are elliptic selfadjoint differential operators.Therefore, if X is compact, their kernels, denoted again with H •,• ∂ (X) and H •,• ∂ (X), are finite dimensional complex vector spaces.Holt and Zhang [11] answered to a question of Kodaira and Spencer [9] showing that, contrarily to the complex case, the dimensions of the spaces of ∂-harmonic (0, 1)-forms on a 4-dimensional manifold depend on the metric.Indeed they construct on the Kodaira-Thurston manifold an almost-complex structure that, with respect to different almost-Hermitian metrics, has varying dim H 0,1 ∂ .With different techniques, in [16] it was shown that also the dimension of the space of ∂-harmonic (1, 1)-forms depends on the metric on 4-dimensional manifolds (for other results in this direction, see [13] and [10]).
We note that performing explicit computations of ∂-harmonic forms is a difficult task and not much is known in higher dimensions (see [15], [3], [4] for some detailed computations).
In the present paper we study the validity of primitive decompositions on compact almost-Kähler manifolds in any dimension.More precisely, in Propositions 3.1 and 3.2, Theorem 3.4 and Corollary 3.5 we prove, on compact almost-Kähler 2n-dimensional manifolds, primitive decompositions for ∂and ∂-harmonic forms in bidegrees (p, 0), (0, q), (1, 1), (n, n−p), (n−q, n) and (n−1, n−1), with p, q ≤ n.One cannot hope to have such decompositions for any bidegree as shown in Example 5.3.For similar results in the case of Bott-Chern harmonic forms, we refer the reader to [12].
We notice that, even though the metric is almost-Kähler, the decompositions of ∂and ∂-harmonic forms might differ.Indeed, in Section 4 we show explicitly that, differently from the Kähler case, one can have ∆ ∂ ̸ = ∆ ∂ , and also We observe that a key ingredient in the proof of the results in [16] (see also [11]) is indeed the primitive decomposition of ∂-harmonic (1, 1)-forms on 4-dimensional manifolds.In fact, in this dimension in Proposition 4.1 we prove the general equality All the examples we present are nilmanifolds, of dimensions 6 and 8, endowed with possibly non-left-invariant almost-Kähler structures.
We recall that if one wants to mimic and recover all the Kähler identities, the proper operator to consider is δ := ∂ + µ (see [5], [14]).However, considering just the operator ∂ on almost-Kähler manifolds we are able to see how genuinely almost-Kähler manifolds differ from Kähler ones.More precisely, the study of the kernel of ∆∂ illuminates the purely almost-complex properties.

Preliminaries
In this section we recall some basic facts about almost-complex and almost-Hermitian manifolds and fix some notations.Let X be a smooth manifold of dimension 2n and let J be an almost-complex structure on X, namely a (1, 1)tensor on X such that J 2 = − id.Then J induces on the space of forms A • (X) a natural bigrading, namely Accordingly, the exterior derivative d splits into four operators: where µ and μ are differential operators that are linear over functions.In particular, they are related to the Nijenhuis tensor N J by where α ∈ A 1 (X).Hence, J is integrable, that is, J induces a complex structure on X if and only if µ = μ = 0.
In general, since d 2 = 0, one has In particular, ∂2 ̸ = 0, and so the Dolbeault cohomology of X is well defined if and only if J is integrable.The same holds for the operator ∂.
If g is an Hermitian metric on (X, J) with fundamental form ω and * is the associated C-linear Hodge- * operator, one can consider the adjoint operators . These spaces will be called the spaces of D-harmonic forms.The operators ∆ ∂ and ∆ ∂ are second-order, elliptic, differential operators; in particular, if X is compact, the associated spaces of harmonic forms are finite-dimensional, and their dimensions will be denoted by h p,q ∂ and h p,q ∂ .If X is compact, then we easily deduce the following relations for a (p, q)-form α: which characterize the spaces of harmonic forms.

Primitive decompositions of Dolbeault harmonic forms
Let (X, J, g, ω) be a 2n-dimensional almost-Hermitian manifold.We denote with the Lefschetz operator and with its dual.A k-form α k on X, for k ≤ n, is said to be primitive if Λα k = 0, or equivalently, L n−k+1 α k = 0.Then, the following vector bundle decomposition holds (see, e.g., [18]): where is the bundle of s-primitive forms.Accordingly, given any k-form α k on X, we can write where Furthermore, the decomposition above is compatible with the bidegree decomposition on the bundle of complex k-forms Λ k C X induced by J, that is, where P p,q X = P k C X ∩ Λ p,q X.For any given β k ∈ P k X, we have the following formula (cf.[18, p. 23 In what follows we will write P • = P • X and so on.We recall that by [5,Corollary 5.4] such decompositions in primitive forms pass to the spaces of d-harmonic forms whenever there exists an almost-Kähler metric.More precisely, if (X, J, ω) is a compact 2n-dimensional almost-Kähler manifold, then, for every p, q, In fact, this holds also for the spaces of δand δ-harmonic forms introduced in [14], where δ := ∂ + µ and δ := ∂ + μ.Indeed, by [14, Proposition 6.2 and Theorem 6.7], one has H p,q d (X) = H p,q δ (X) = H p,q δ (X).Next, we are going to study such decompositions for ∂-harmonic forms.First, notice that, since (p, 0)-forms and (0, q)-forms are trivially primitive, we immediately derive the following results.Proposition 3.1.Let (X, J, g, ω) be a compact 2n-dimensional almost-Hermitian manifold (with n ≥ 2).Then the following decompositions hold for every p, q ≤ n: By applying to such decompositions the Hodge- * operator and formula (3.2), we obtain the following result.Proposition 3.2.Let (X, J, g, ω) be a compact 2n-dimensional almost-Hermitian manifold (with n ≥ 2).Then the following decompositions hold for every p, q ≤ n: As a consequence, we derive the following corollary.
Corollary 3.3.Let (X, J, g, ω) be a compact 2n-dimensional almost-Hermitian manifold (with n ≥ 2).Then, Proof.This follows taking p = n and q = n in Proposition 3.2.Otherwise, it can be proved directly.Indeed, let α be an (n, 0)-form (the case (0, n) is similar); then α is primitive, and by Formula (3.2), * α = c n α, with c n ̸ = 0 a constant depending only on the dimension of X.Therefore, for bidegree reasons, We show now that primitive decompositions hold also in other suitable degrees as soon as we assume the existence of an almost-Kähler metric.Theorem 3.4.Let (X, J, g, ω) be a compact 2n-dimensional almost-Kähler manifold (with n ≥ 2).Then the following decomposition holds: Then the primitive decomposition (3.1) reads as where Therefore, by (3.3), (3.5), taking into account that g is almost-Kähler, equations (3.4) are equivalent to After multiplying the first equation by ω n−2 and the second by (n − 2)!, we obtain and taking the sum of the last two equations we obtain By definition, we have Applying −id c to both sides of the above equation, we obtain since ∂ ∂ + ∂∂ = 0 on functions and the other contributions vanish by bidegree reasons when we take the wedge product with ω n−1 .Therefore, from which we derive that β ≡ β 0 ∈ C is constant (see, for instance [8], [1,Theorem 10] or [16,Proposition 3.4] for the 4-dimensional case).Hence we have that β ∈ H 1,1 ∂ and β 1,1 is primitive.This proves that , we easily conclude that ∂ * α 1,1 = 0 and ∂α 1,1 = 0.The decomposition is thus proved.□

A. CATTANEO, N. TARDINI, AND A. TOMASSINI
As a consequence we obtain the following primitive decompositions.
Corollary 3.5.Let (X, J, g, ω) be a compact 2n-dimensional almost-Kähler manifold (with n ≥ 2).Then the following decompositions hold: Proof.The first decomposition follows from the one proved in Theorem 3.4 by conjugation.
To prove the second, observe that the Hodge- * operator induces an isomorphism . Via this isomorphism, ω corresponds to ω n−1 , while by (3.2) on primitive (1, 1)-forms we have * = − 1 (n−2)!L n−2 .So we just have to apply * to the decomposition of the previous point.
Finally, the last point follows from the second by conjugation.□ Recall that by [5] (see also [14]) on compact almost-Kähler manifolds we have and so, for every p, q, In particular, if J is integrable, namely (X, J, g, ω) is a compact Kähler manifold, one recovers the well-known identities ∂ .Therefore, one could wonder if this last identity holds true also in the non-integrable case for some special bidegrees.More precisely, we want to show that the two primitive decompositions we obtained in Theorem 3.4 and Corollary 3.
We show now that in higher dimension the equality does not hold in general.
Example 4.2.Let T 6 = Z 6 \R 6 be the 6-dimensional torus with coordinates (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ) on R 6 .Let f = f (x 2 ) be a non-constant Z-periodic function, and we define the following non-left-invariant almost-complex structure J on A global co-frame of (1, 0)-forms is given by The structure equations are and dΦ 2 = dΦ 3 = 0.Then, the (1, 1)-form is a compatible symplectic structure, namely (J, ω) is an almost-Kähler structure on T 6 .Notice now that by a direct computation Therefore, from [5], we have The last point follows either by direct computation or by noticing that μ * μΦ Another example is provided by the following 8-dimensional nilmanifold with a left-invariant almost-Kähler structure.

By direct computation we get
hence Therefore, on (1, 1)-forms.However, one can show that their kernels coincide, namely Remark 4.4.We want to point out that finding explicit examples of almost-Kähler manifolds with ∆∂ ̸ = ∆ ∂ seems to be not so obvious.In fact, we couldn't find any left-invariant example in dimension 6.

Primitive decompositions in dimension 6
Notice that in view of Propositions 3.1, 3.2, Theorem 3.4 and Corollary 3.5 we have a full primitive description of all ∂-harmonic forms on compact 4-dimensional almost-Kähler manifolds.It is therefore natural to ask what happens for bidegrees different from (p, 0), (0, q), (n, q), (p, n), (1, 1) and (n−1, n−1) in higher dimension.The first interesting dimension to consider is 6, and in this case the only bidegrees left are (2, 1) and (1, 2).Let us focus, for instance, on bidegree (2, 1).The primitive decomposition of forms is Passing to ∂-harmonic forms, it follows that indeed, on compact almost-Kähler manifolds, for bidegree reasons and [5] one has μ .Therefore, it is natural to wonder whether such inclusion is indeed an identity.In fact, this is not the case in general, as shown by the following proposition.

Proposition 5.1. There exists a compact almost-Kähler
Proof.We refer the reader to Example 5.3 for the proof.□ First we need the following lemma, which will allow us to work only with leftinvariant forms.Lemma 5.2.Let X 6 = Γ\G be the compact quotient of a 6-dimensional, connected, simply-connected Lie group by a lattice and let (J, ω) be a left-invariant almost-Kähler structure on X.Let η ∈ A 2,1 (X) be a left-invariant (2, 1)-form on X with primitive decomposition η = α + Lβ.Then, α and β are left-invariant.