Dolbeault and $J$-invariant cohomologies on almost complex manifolds

In this paper we relate the cohomology of $J$-invariant forms to the Dolbeault cohomology of an almost complex manifold. We find necessary and sufficient condition for the inclusion of the former into the latter to be true up to isomorphism. We also extend some results obtained by J. Cirici and S.O. Wilson about the computation of the left-invariant cohomology of nilmanifolds to the setting of solvmanifolds. Several examples are given.


Introduction
Let (M, J) be a 2m-dimensional almost complex manifold. Then the almost complex structure J induces a bigrading on the bundle of differential forms on M . The exterior derivative d acts on differential forms as the sum of four differential operators, d = µ + ∂ +∂ +μ. The celebrated theorem of Newlander and Nirenberg states that M admits the structure of complex manifold, i.e., J is integrable, if and only if N J = 0, that is equivalent to µ =μ = 0. Consequently, in such a case d = ∂ +∂. For complex manifolds it is classical and well established the theory of Dolbeault cohomology, obtained as the cohomology of the∂ operator. Another fundamental tool is the Hodge Theory for the∂ operator that, once fixed a Hermitian metric, establishes an isomorphism between the Dolbeault cohomology and the kernel of the Dolbeault Laplacian ∆∂. However for almost complex manifolds, the operator∂ is not cohomological and the Dolbeault cohomology cannot have the usual definition. It is natural to look for other cohomological theories to study geometric properties of almost complex manifolds. Motivated by the comparison between the J-tamed symplectic cone K t J and the J-compatible symplectic cone K c J of an almost complex manifold, defined as the projection in cohomology of the space of symplectic forms taming J, respectively calibrating J, Li and Zhang introduced in [LZ09] the J-invariant cohomology, respectively J-anti-invariant cohomology groups of an almost complex manifold (M, J), denoted with H + , respectively H − , formed by 2 nd -de Rham classes represented by closed J-invariant, respectively J-anti-invariant forms, with respect to the natural action of J on the space of 2-forms. Such groups generalize the real Dolbeault cohomology classes in H 1,1 ∂ ∩ H 2 dR (R) and H 2,0 ∂ + H 0,2 ∂ ∩ H 2 dR (R) respectively. The focus is on whether the almost complex structure J is C ∞ -pure, i.e., H + ∩H − = {0} or C ∞ -full, i.e., H 2 dR = H + +H − . The problem is further studied in [DLZ10], where it is proved that any almost complex structure on a compact 4-manifold is C ∞ -pure and C ∞ -full, and in [DLZ11]. Such a result can be viewed as a sort of Hodge decomposition for 4-dimensional compact almost complex manifolds.
Recently J. Cirici and S. O. Wilson defined in [CW18a] an analogous of Dolbeault cohomology for almost complex manifolds, that is also called Dolbeault cohomology. This idea of cohomology is based on the decomposition of d and allows a development of a harmonic theory, at least in some favorable situation such as in [CW18b] for the almost Kähler case (see also [TT20]). A Frölicher spectral sequence E p,q r builds a bridge between the Dolbeault cohomology and the complex de Rham cohomology. In general, the computation of such groups is difficult, since they might not be finite-dimensional. A special setting in which calculations can be performed is that of Lie Algebra. Such computations have a direct application in the study of the left-invariant Dolbeault cohomology of nilmanifolds, as showed in [CW18a].
In this paper we study the relation between the complex cohomology group H + C of J-invariant complex forms and the Dolbeault cohomology group H 1,1 Dol on almost complex manifolds. Next we extend some results obtained in [CW18a] for nilmanifolds, to the case of solvmanifolds. More in details, since we have a characterization of J-invariant 2-forms as real forms of complex bidegree (1, 1), it is natural to ask whether they belong or not to the Dolbeault cohomology groups, or at least if there exists an isomorphism between H + and a subgroup of H 1,1 Dol . We relate J-invariant cohomology and Dolbeault cohomology, finding that the condition is necessary and sufficient for the former cohomology group to be contained into the latter up to isomorphism (Theorem 4.1). Then given any solvmanifold endowed with a left-invariant almost complex structure, we prove that the left-invariant spectral sequence satisfies Serre duality at every stage and that the left-invariant Dolbeault cohomology groups are isomorphic to the kernel of a suitable Laplacian (Theorem 6.1). Finally calculations of left-invariant spectral sequence and J-invariant cohomology are performed on almost complex manifolds and solvmanifolds endowed with a left-invariant almost complex structure to give concrete applications. The paper is organized as follows. In section 2 we briefly recall some basic definitions that will be used later on, and we introduce the notation. In section 3 we resume the definition given by Cirici and Wilson of Dolbeault cohomology for almost complex manifolds. In particular, we focus on the spectral sequence arising from a Hodge filtration, and give an explicit description of it. Section 4 is devoted to the study of J-invariant cohomology and Dolbeault cohomology. We prove the results mentioned above, and investigate the behaviour of the necessary and sufficient condition under small deformations, proving with an example that it is not a closed property. Section 5 recalls the construction of the Dolbeault cohomology of Lie Algebras, while in section 6 we prove the Serre duality for solvmanifolds. Finally in section 7 we collect various examples of Dolbeault cohomology and spectral sequence. Among them, we provide computations of the left-invariant spectral sequence on 4-dimensional solvmanifolds that do not admit any integrable almost complex structure. For such examples the Dolbeault cohomology theory for almost complex manifolds becomes the main tool to investigate their geometry.
Acknowledgements. The authors would like to thank Joana Cirici and Weiyi Zhang for useful comments and remarks.

Preliminaries and notation
Let (M, J) be an almost complex manifold of real dimension 2m, with J an almost complex structure on the tangent bundle, i.e., J ∈ End(T M ) such that J 2 = −Id. Denote by A * R (respectively A * C ) the algebras of real (respectively complex) differential forms on M . J induces a bigrading on complex forms, On real k-forms, α ∈ A k C , J induces a map still denoted by J and defined as (2.2) Jα(X 1 , . . . , X k ) = α(JX 1 , . . . , JX k ).
If k is odd, J 2 = −Id, while if k is even, J is an involution. In particular, A 2 R decomposes as R denotes the J-invariant forms and A − R the J-anti-invariant forms. If we consider the bidegree induced on complex forms by J, it's easy to check that A + R consists of real forms in A 1,1 C , while A − R of real forms in A 2,0 C + A 0,2 C . We denote with H * dR (R) (respectively H * dR (C)) the real (respectively complex) de Rham cohomology of M . The de Rham cohomology groups consisting of J-invariant and J-anti-invariant forms were introduced in [LZ09]. We shall use the notation of [DLZ11]. The Jinvariant real cohomology group is In the following we will denote a (p, q)-form α with α p,q .
We call a solvmanifold the quotient of a connected, simply connected and solvable Lie Group G, by a discrete and co-compact subgroup Γ of G. We denote it by Γ\G. If G is also nilpotent, we call Γ\G a nilmanifold.
In general,∂ does not square to 0 on M , and its cohomology is not well defined. From the relation∂μ +μ∂ = 0,∂ is well defined on cohomology classes of H p,q µ , and thanks to∂ 2 +μ∂ + ∂μ = 0, it squares to 0, thus we can define the Dolbeault cohomology of the almost complex manifold (M, J) as the∂-cohomology of thē µ-cohomology, i.e., .
We say that the spectral sequence degenerates at stage r, for bidegree (p, q), and write E p,q The spectral sequence degenerates at stage r if it degenerates at stage r for all bidegrees.
At the E ∞ stage, the degeneration of the spectral sequence induces a bigrading on the de Rham cohomology of the almost complex manifold. In particular, E p,q ∞ represents cohomology classes in H p+q dR (C) that admit a complex representative of bidegree (p, q).

Inclusion of J -Invariant cohomology into Dolbeault cohomology
Denote by H + C = H + ⊗ C the complexified of the J-invariant cohomology group. We are going to study under which conditions H + C is isomorphic to a subgroup of H 1,1 Dol through the isomorphism defined in (3.9) between the Frölicher spectral sequence and the quotients X p,q r /Y p,q r . The results are stated in Theorem 4.1, that gives a characterization valid in the almost complex case. At the end, we briefly investigate the stability of the condition under small deformations (in the integrable case).
For the inclusion up to isomorphism, the key condition is the isomorphism between two terms of the spectral sequence ( * ) E 0,1 1 = E 0,1 2 , thus we find meaningful considering the openness and closedness of ( * ) under small deformations of complex structure. Let M be a compact complex manifold and {J t } a deformation of complex structures on M , with small t ∈ C. An easy calculation shows that if we assume ( * ) for t = 0, the function e 0,1 . We ask the following Question: let (M, J 0 ) be a compact complex manifold. Is condition ( * ) stable under small deformations of the complex structure J 0 ?
In all the examples for which we performed computations, the stability is satisfied. On the other side, as a consequence of Example 7.1, we have that ( * ) is not a closed condition. More precisely, the example shows the following proposition.
Proposition 4.2. There exist complex manifolds (M, J) such that (i) the spectral sequence satisfies The same is true at left-invariant level for almost complex manifold as shown in Example 7.4.

Dolbeault cohomology of Lie Algebras
Let g be a real Lie Algebra of dimension 2m and J a complex structure on the vector space g. We call J an almost complex structure on the Lie Algebra g. Consider the Chevalley-Eilenberg complex of g, (A * g , d). Recall that the differential is defined as the dual of the Lie bracket [·, ·] for 1-forms, and extended as a derivation to all forms. J induces a bidegree on the complexified of the Chevalley-Eilenberg complex, The Dolbeault cohomology of the Lie Algebra g is the∂-cohomology of theμcohomology groups, , and the spectral sequence {E * , * r (g)} r∈N associated to the Hodge filtration is the spectral sequence of g. In the setting of Lie Algebras, it's possible to compute easily the cohomology as a matter of linear algebra and all the spaces are finite dimensional. We set , and . As a consequence of existence of the spectral sequence, we have Frölicher inequalities (cf. [Frö55]) for the almost complex case.
Theorem 5.1 ([CW18a] Proposition 5.1). Let g be a real Lie Algebra, dim g = 2m and J an almost complex structure on g. Then Denote by χ(g) = k (−1) k b k the Euler characteristic of g. Then Consider now a J-compatible inner product ·, · on g. It is possible to develop an harmonic theory for differential operators that makes easier some computations of cohomology groups. The Hodge * operator is defined as usual by the relation with V ol denoting the volume form in A 2m g C , and ϕ, η ∈ A p,q g C . Taken δ among d, µ, ∂,∂,μ, the formal adjoint of δ is the operator (5.9) δ * = − * δ * .
The δ-Laplacian is defined as (5.10) ∆ δ = δδ * + δ * δ, and the space of δ-harmonic (p, q)-forms is On a Lie Algebra, the above spaces are always finite dimensional. The operatorμ * is the adjoint ofμ with respect to ·, · , and A p,q g C admits a Hodge decomposition In particular, cohomology classes in H p,q µ (g) admit aμ-harmonic representative. For Lie Algebras, the Dolbeault cohomology can also be obtained as the cohomology of the operator∂μ, defined onμ-harmonic (p, q)-forms as (5.13)∂μ(ϕ) = Hμ(∂ϕ), where we denoted with Hμ the projection onμ-harmonic forms. It can be checked that∂μ is a cohomological operator and (5.14) .

Dolbeault cohomology of solvmanifolds endowed with a left-invariant almost complex structure
In this section we extend results obtained from J. Cirici and S. O. Wilson in [CW18a] for nilmanifolds, to the case of solvmanifolds, showing that the left-invariant Dolbeault cohomology always satisfies Serre duality and is described by∂μ.
Let M = Γ\G be a solvmanifold. There is a natural left action G × M −→ M . Consider the three graded algebra: • A * g , algebra of differential forms on g; Let J be an almost complex structure on the Lie Algebra g. Then via the isomorphism (6.1), the bigrading of A * , * g C induces a bigrading on the left-invariant complex forms, that corresponds to a left-invariant almost complex structure (still denoted by J), and vice-versa, every left-invariant almost complex structure induces an almost complex structure on J. J is integrable on g if and only if it is integrable as an almost complex structure on left-invariant forms. We extend J to non-invariant vector fields by linearity over functions. The extendedJ is integrable iff its Nijenhuis tensor NJ vanishes iff N J vanishes iff J is integrable on g.
The left-invariant Dolbeault cohomology of M is defined as the Dolbeault cohomology of the complexified Lie Algebra, (6.5) L H p,q Dol (M ) = H p,q Dol (g), and the left-invariant spectral sequence of M as the spectral sequence associated to the Dolbeault cohomology of g, (6.6) L E * , * r = E * , * r (g). M also admits a Dolbeault cohomology as an almost complex manifold (M,J). It is not known if this cohomology coincides with the left-invariant one, even in the case when (6.4) holds, but this is conjectured to be true for nilmanifolds and integrable almost complex structures (cf. [Rol11]). We want to prove the following theorem.
Theorem 6.1. Let M = Γ\G be a solvmanifold. Then for all (p, q), its leftinvariant Dolbeault cohomology is obtained as∂μ-harmonic forms, The left-invariant spectral sequence satisfies Serre duality at every stage Before giving the proof of the theorem, we make some preliminary observation and state some useful Lemma.
The study of left-invariant cohomology is made easy if∂ * µ is the metric adjoint of ∂μ. A sufficient condition for this to happen (cf. [CW18a] Lemma 5.2) is Equivalent conditions to (6.9) are (6.10) d ≡ 0 on A 2m−1 g C , and (6.11) H 2m dR (g; C) ∼ = C. We recall now some consequences of (6.10), that will be used to prove Theorem 6.1. The fact that∂ * µ is adjoint of∂μ, allows to use harmonic theory to establish an isomorphism from∂μ-harmonic forms to the Dolbeault cohomology of g, (and consequently to the left-invariant cohomology of M ).  and∂μ-harmonic forms are isomorphic to the Dolbeault cohomology of g, (6.13) H p,q Dol (g) ∼ = H p,q ∂μ . The Hodge * operator and conjugation give, with the usual argumentation, Serre duality for H p,q ∂μ , and via the above isomorphism, the first stage of the left-invariant spectral sequence also satisfies Serre duality. For the sake of completeness, we recall the proof of the following well known result.
Lemma 6.4. Let g be an unimodular Lie Algebra, dim R g = 2m. Then d ≡ 0 on Proof. Let {e j } 2m j=1 be a basis of g, and {φ j } 2m j=1 a dual basis. Set [e j , e k ] = l C l jk e l , C l jk + C l kj = 0 The differential on 1-forms is determined by the structure constants A basis of (2m − 1)-forms is given by {φ 1 ∧ · · · ∧φ j ∧ · · · ∧ φ 2m } 2m j=1 , whereφ j means that the form is omitted. Then we have where V ol = φ 1 ∧ · · · ∧ φ 2m , and the last equality follows by definition of trace If g is unimodular, the trace of the adjoint vanishes.
We are ready to proceed with the proof.
Proof of Theorem 6.1. Since G is a connected, simply connected solvable Lie Group that admits a lattice, it is unimodular and by Lemma 6.4, condition (6.10) is satisfied. Proposition 6.2 and 6.3 are valid for g and so the left-invariant Dolbeault cohomology group of M are isomorphic to left-invariant∂μ-harmonic forms and satisfy Serre duality. This proves (6.7), and also (6.8) for the first stage. For r = 1, note that ( L E * , * 1 , d 1 ) satisfies the hypothesis of the main theorem in [Mil20]. In fact we proved that d vanishes on A 2m−1 g C , thus also ∂,∂ andμ vanish on A 2m−1 g C . d 1 is a sum of such differentials, thus d 1 = 0. Serre duality at first stage, d 1 = 0 on A 2m−1 g C and L E m,m 1 ∼ = C imply Serre duality at every stage.

Examples
In this section we show some example of what we proved in section 4 and in section 6. We begin showing how, on a complex manifold, that ( * ) is not a closed condition. We also note that when it is not satisfied, cohomology classes in H + C do not define cohomology classes in H 1,1 ∂ .
The basis of (1, 0)-forms parametrized by t, has the following differentials: t . If |sin t| = 1, we have that the first stage of the spectral sequence coincides with the second one (but not with the third one). In particular, condition ( * ) is satisfied. For sin t = 1, we have thus H 0,1 ∂ ∼ = C 3 , while E 0,1 2 ∼ = C 2 and ( * ) is not satisfied.
Note that d(ϕ3 t + ϕ3 t ) = 1 2 (ϕ 12 t + ϕ 21 t ) = ∂ϕ3 t +∂ϕ 3 t is a d-exact real 2-form and thus belongs to the 0 class in H + C . However, it does not belong to the 0 class in the Dolbeault cohomology of (1, 1)-forms since is not the∂ of any (1, 0)-form, and the inclusions H + ⊆ H 1,1 ∂ and so H + C ⊆ H 1,1 ∂ are not well defined.
The second example comes from the holomorphically parallelizable Nakamura manifold, and its small deformations. The manifold can be obtained as the quotient of the Lie Group G = C ⋉ φ C 2 , with by the lattice Γ = a + ib, c + id . If we assume that b, d ∈ πZ (cf. [AK17a], Example 3.4), we can compute the Dolbeault cohomology of the manifold using the (0, 1)-forms dz1, e z1 dz2, e z1 dz3, and the conjugate (1, 0)-forms. The deformation we are interested in, seen as an Direct calculations show that for all t = 0, the spectral sequence degenerates at the first stage. For t = 0, we have H 0,1 ∂ ∼ = C 3 and E 0,1 2 ∼ = C, thus ( * ) is not satisfied.
This fact is resumed in [AK17b], Proposition 4.2. The above manifolds, both provide a proof of Proposition 4.2.
We proceed now to compute the Dolbeault left-invariant cohomology of 4-dimensional solvmanifolds that do not admit integrable almost complex structures. There exists three such solvmanifolds (cf. [Boc16]). The first one is a nilmanifold. The second one is a symplectic, completely solvable but not nilpotent solvmanifold (Example 7.2) and the last one is a completely solvable but not symplectic nor nilpotent solvmanifold (Example 7.5). Calculations for the nilmanifold have already been made in [CW18a], Example 5.15.
Example 7.2 (Γ\Sol(3) × S 1 ). Denote by Sol(3) the solvable Lie Group of dimension 3. It can be obtained considering the groups (R, +), (R 2 , +) and taking the semi-direct product Sol(3) = R ⋉ φ R 2 , with The product Sol(3) × R can be identified as a subgroup of matrices via the homomorphism Denote with K the image of Sol(3) × R by θ. Then K is a subgroup of SL(5, R) with respect to matrix multiplication, isomorphic to Sol(3) × R. By [AGH61] (Theorem 4), K admits a lattice Γ. The quotient M = Γ\K is a solvmanifold, of real dimension 4. An explicit construction of M can be found in [Boc16] in the examples following the classification of four-dimensional solvmanifolds. Taking A ∈ K and computing A −1 dA, we obtain a basis of left-invariant forms e 1 = dt, e 2 = e −t dx, e 3 = e t dy, e 4 = ds .
In both examples, harmonic representatives of H + C are also harmonic representative of L H 1,1 Dol . For structure (A), the condition L E 0,1 1 = L E 0,1 2 is satisfied, and if we consider a non-harmonic representative in H + C , it still defines a class in L H 1,1 Dol , as expected from Theorem ??. In fact we have H + C = ϕ1 2 − ϕ 12 . The form is d-closed. d-exact (1, 1) forms are written as∂β 1,0 + ∂β 0,1 , with with the conditionsμβ 1,0 +∂β 0,1 and µβ 0,1 +∂β 1,0 that are satisfied only if b+d = 0. Immediately we have∂ that is the 0 class in L H 1,1 Dol . This is not true for structure (C), in fact if we modify ϕ 11 with a d-exact (1, 1)-form, the class in L H 1,1 Dol varies. This last condition is written as where A and B are 2 × 2 matrices and P = 0 1 1 0 .
The deformations are also codified by a form ψ ∈ T 1,0 M ⊗ T 0,1 M * that is written as ψ = ψ 1 1 ϕ1 ⊗ Z 1 + ψ 2 1 ϕ1 ⊗ Z 2 + ψ 1 2 ϕ2 ⊗ Z 1 + ψ 2 2 ϕ2 ⊗ Z 2 , and must satisfy ψ = 1 2 (L − iJL). By writing out both members of the equality we obtain the expression of ψ in function of A and B, then we compute the brackets of the deformed structures in function of ψ and the brackets at time 0. Finally by duality we obtain the differentials of the deformed left-invariant forms, and compute the left-invariant spectral sequence. We classify deformations into two groups: (i) A 21 + A 12 = 0 and B 11 = 0, (ii) the remaining structures.
For structures of type (i), the behaviour of the spectral sequence stays the same, and we have degeneracy at the second stage. For structures of type (ii) and t = 0, we have that the spectral sequence degenerates at the first stage and coincides with the spectral sequence of structure A and B. In particular this shows that condition L E 0,1 1 = L E 0,1 2 is not closed (at level of left-invariant spectral sequence), since it is not satisfied for t = 0, but it is true for deformations of class (ii). The proof that G admits a lattice Γ, and an explicit construction of the quotient can be found, as for example 7.2, in [Boc16]. Then the quotient M = Γ\G is a solvmanifold, of real dimension 4. A basis of left-invariant forms is e 1 = dt, e 2 = e −α2t dx, e 3 = e −α3t dy, e 4 = e −α4t dz .