Bott-Chern Laplacian on almost Hermitian manifolds

Let $(M,J,g,\omega)$ be a $2n$-dimensional almost Hermitian manifold. We extend the definition of the Bott-Chern Laplacian on $(M,J,g,\omega)$, proving that it is still elliptic. On a compact K\"ahler manifold, the kernels of the Dolbeault Laplacian and of the Bott-Chern Laplacian coincide. We show that such a property does not hold when $(M,J,g,\omega)$ is a compact almost K\"ahler manifold, providing an explicit almost K\"ahler structure on the Kodaira-Thurston manifold. Furthermore, if $(M,J,g,\omega)$ is a connected compact almost Hermitian $4$-manifold, denoting by $h^{1,1}_{BC}$ the dimension of the space of Bott-Chern harmonic $(1,1)$-forms, we prove that either $h^{1,1}_{BC}=b^-$ or $h^{1,1}_{BC}=b^-+1$. In particular, if $g$ is almost K\"ahler, then $h^{1,1}_{BC}=b^-+1$, extending the result by Holt and Zhang for the kernel of Dolbeault Laplacian. We also show that the dimensions of the spaces of Bott-Chern and Dolbeault harmonic $(1,1)$-forms behave differently on almost complex 4-manifolds endowed with strictly locally conformally almost K\"ahler metrics. Finally, we relate some spaces of Bott-Chern harmonic forms to the Bott-Chern cohomology groups for almost complex manifolds, recently introduced by Coelho, Placini and Stelzig.


Introduction
On a complex manifold, given a Hermitian metric, several elliptic operators naturally arise from the union of the complex and the Hermitian structure. As a typical example, the Dolbeault Laplacian is defined as ∆ ∂ = ∂∂ * + ∂ * ∂, where the exterior differential defined on the space A p,q of (p, q)-forms decomposes as d = ∂ +∂ and, if * ∶ A p,q → A n−q,n−p is the C-linear complex Hodge star operator, where n is the complex dimension of the complex manifold, then ∂ * = − * ∂ * and ∂ Schweitzer, in 2007, [17], studied the operator∆ BC on compact Hermitian manifolds, naming it the Bott-Chern Laplacian. In particular, denoting by H p,q BC the space of Bott-Chern harmonic (p, q)-forms on a given compact Hermitian manifold (M, J, g, ω), he proved the following Bott-Chern decomposition of the space of (p, q)-forms As a consequence, the space H p,q BC is finite dimensional and H p,q BC ≅ H p,q BC , where H p,q BC = ker d im ∂∂ denotes the (p, q)-Bott-Chern cohomology group. In particular, the complex dimension h p,q BC = dim C H p,q BC is a complex invariant of (M, J), which does not depend on the Hermitian metric g.
If the compact Hermitian manifold (M, J, g, ω) is Kähler, i.e., dω = 0, then and the spaces of Bott-Chern and Dolbeault harmonic forms coincide, i.e., Now, let (M, J, g, ω) be an almost Hermitian manifold, i.e., the almost complex structure J may not be integrable, i.e., J may not derive from a complex-manifold structure on M . The exterior differential decomposes as d = µ + ∂ + ∂ + µ, and Dolbeault and Bott-Chern cohomologies are, in general, no more well defined. However, the Dolbeault Laplacian ∆ ∂ is still well defined and elliptic, resulting in H p,q ∂ being finite dimensional when M is compact.
The study of Dolbeault harmonic forms on almost Hermitian manifold of real dimension 4 has been very recentely developed by Holt and Zhang, [12,13], and by Tardini and the second author, [19]. Holt and Zhang, working on the Kodaira-Thurston manifold, showed that the number h 0,1 ∂ may become arbitrarily large when varying continuously the almost complex structure with an associated almost Kähler metric and that h 0,1 ∂ may vary with different choices of almost Hermitian metrics. Furthermore, they proved that h 0,1 ∂ may vary with different choices of almost Kähler metrics. In this way they answered a question by Kodaira and Spencer, [10,Problem 20]. Moreover, they showed h 1,1 ∂ = b − + 1 on every compact almost Kähler 4-manifold, where b − is the dimension of the space of anti-self-dual, i.e., * α = −α, Hodge harmonic 2-forms, which is a topological invariant. Tardini and the second author proved that h 1,1 ∂ = b − on every compact almost complex 4-manifold with a strictly locally conformally almost Kähler metric.
In this paper, we focus on the study of the Bott-Chern Laplacian on almost Hermitian manifolds. Note that, analogously to the Dolbeault Laplacian, also the Bott-Chern Laplacian∆ BC is still well defined on almost Hermitian manifolds (M, J, g, ω), and it is straightforward to show that it is also elliptic, see Proposition 3.2. Therefore, when M is compact, the Bott-Chern decomposition (1) still holds, and H p,q BC is finite dimensional. We prove the following Theorem (Theorem 4.3). Let (M, J, g, ω) be a compact almost Hermitian manifold of real dimension 4. Then either h 1,1 BC = b − or h 1,1 BC = b − + 1. Moreover, we specialize the previous theorem when the almost Hermitian metric ω is almost Kähler, i.e., dω = 0, obtaining that h 1,1 BC is independent of the choice of an almost Kähler metric on a given compact almost complex 4-manifold, that is, Theorem (Corollary 4.4). Let (M, J, g, ω) be a compact almost Kähler manifold of real dimension 4. Then, h 1,1 Note that in the integrable case, i.e., on compact complex surfaces, it holds that h 1,1 BC = b − +1 on Kähler surfaces, on complex surfaces diffeomorphic to solvmanifolds, and on complex surfaces of class VII (see [2, Chapter IV, Theorem 2.7] and [1]).
We also provide a non integrable almost complex structure on a hyperelliptic surface, endowed with a strictly locally conformally almost Kähler metric, such that h 1,1 BC = b − + 1. This proves that the dimension of Bott-Chern harmonic (1, 1)-forms behaves differently than the dimension of Dolbeault harmonic (1, 1)-forms, [19], when the almost complex 4-manifold is endowed with a strictly locally conformally almost Kähler metric.
Very recently Holt improved the result of Theorem 4.3, by showing that h 1,1 BC = b − + 1 on any given compact almost Hermitian 4-manifold, see [11,Theorem 4.2].
Taking into account the integrable case, one may ask whether (2) and (3) holds or not, when the almost Hermitian metric is almost Kähler. We show that (3) is not true, describing an explicit example on the Kodaira-Thurston manifold. This also implies that (2) does not hold. In fact, working on a family of almost Kähler metrics on the Kodaira-Thurston manifold, we show Theorem (Corollary 5.3). There exists an almost Kähler 4-manifold (M, J, g, ω) such that for some bidegree (p, q) it holds that Finally, we recall a very recent definition of Bott-Chern cohomology for almost complex manifolds, [5], obtaining a natural injection of some spaces of Bott-Chern harmonic forms into this new Bott-Chern cohomology.
For other results concerning Bott-Chern-like harmonic forms on almost complex manifolds, equipped with cohomological counterparts, see [18].
The present paper is organized in the following way. In section 2, we review some basic facts on almost complex manifolds and elliptic differential operators. Section 3 is devoted to the definition and to the proof of the fundamental properties of the Bott-Chern Laplacian in the almost complex setting. In section 4, we study Bott-Chern harmonic (1, 1)-forms on almost Hermitian 4-manifolds, proving Theorem 4.3. In section 5, we describe a family of almost Kähler structures on the Kodaira-Thurston manifold, comparing the two spaces H p,q BC and H p,q ∂ and proving Corollary 5.3. In section 6, we describe an almost complex structure on a hyperelliptic surface, endowed with a strictly locally conformally almost Kähler metric, such that h 1,1 BC = b − + 1. Finally, in section 7, we recall a very recent definition of Bott-Chern cohomology for almost complex manifolds by Coelho, Placini and Stelzig [5], and briefly analyse its relation with the space of Bott-Chern harmonic forms (see Proposition 7.1).

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 2ndifferentiable manifold together 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 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 = A p,q (M ) = Γ(Λ p,q M ) be the space of smooth sections of Λ p,q M . We denote by Let f ∈ C ∞ (M, C) 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 differential acts as Therefore, it turns out that the differential operates on (p, q)-forms as where we denote the four components of d by Let (M, J) be an almost complex manifold. If the almost complex structure J is induced from a complex manifold structure on M , then J is called integrable. It is equivalent to the decomposition of the exterior differential as d = ∂ + ∂.
A Riemannian metric on M for which J is an isometry 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, g, 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 T 1,0 M × T 1,0 M → C and g as a C-bilinear operator T 1,0 M × T 0,1 M → C. Recall that h(u, v) = g(u,v) for all u, v ∈ Γ(T 1,0 M ).
Let (M, J, g, ω) be an almost Hermitian manifold of real dimension 2n. Extend h on (p, q)-forms and denote the Hermitian inner product by ⟨⋅, ⋅⟩. Let * ∶ A p,q (M ) → A n−q,n−p (M ) 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 Then the operators are the formal adjoint operators respectively of d, µ, ∂, ∂, µ. Recall ∆ d = dd * + d * d is the Hodge Laplacian, and, as in the integrable case, set respectively as the ∂ and ∂ Laplacians.
If M is compact, then we easily deduce the following relations . Since we will need to use the maximum principle for second order uniformly elliptic differential operators, let us recall some definitions and the results which will be useful. Let M be a differentiable manifold of dimension m, and let E, F be K-vector bundles over are s × r matrices with smooth coefficients on Ω.
Let P ∶ Γ(M, E) → Γ(M, F ) be a K-linear differential operator of order l from E to F . We define the principal symbol of P as the operator is a smooth section of E and f ∈ C ∞ (M ) is a smooth real valued function, then and consider a Riemannian or a Hermitian metric g on E. We say that P is strongly elliptic if l = 2k and there exists C > 0 such that for all x ∈ M , u ∈ Γ(M, E) and ξ ∈ T * M , see [15,Definition 4.2]. We will make use of the following statement of the maximum principle for strongly elliptic operators of second order, see [7, Chapter 6, Section 4, Theorem 3].
Theorem 2.1. Let Ω ⊂ M be a relatively compact domain, with Ω contained in a local chart, and let P ∶ C ∞ (Ω) → C ∞ (Ω) be a strongly elliptic R-linear differential operator of order 2 without zero order terms, i.e., such that P (1) = 0. If P u = 0 in Ω and u ∈ C(Ω) attains its maximum or minimum over Ω at an interior point, then u is constant within Ω.

Bott-Chern and Aeppli Laplacians
Let (M, J, g, ω) be an almost Hermitian manifold. As in the integrable setting, we define∆ and still call them the Bott-Chern and the Aeppli Laplacian, respectively. Note that If M is compact, then we easily deduce the following relations which characterize the spaces of harmonic (p, q)-forms H p,q BC , H p,q A , defined as the spaces of (p, q)-forms which are in the kernel of the associated Laplacians.
Remark 3.1. By equation (5), note that * H p,q BC = H n−q,n−p A and * H p,q A = H n−q,n−p BC . In the following, we will study only the spaces H p,q BC on an almost complex manifolds; this is sufficient to describe also the spaces H p,q A . We are interested in studying the kernel of the Bott-Chern Laplacian∆ BC on almost complex manifolds. The kernel of an elliptic operator is finite dimensional on a compact manifold. Therefore, the first thing we verify is that∆ BC is elliptic. The proofs known by the authors of the ellipticity of∆ BC , see, e.g., [16,Proposition 5] by Kodaira and Spencer or [17,Page 8] by Schweitzer, make use of local complex coordinates to compute explicitly the symbol of∆ BC , therefore do not hold anymore on almost complex manifolds. Nonetheless, these proofs could be adapted to compute the symbol in suitable local frames on almost complex manifolds. Proof. To compute the symbol of∆ BC , choose a local coframe {θ 1 , . . . , θ n } on A 1,0 such that the almost Hermitian metric is written We write a form α ∈ A p,q locally as Its differential then acts as In calculating the symbol, we are only interested in the highest order derivatives acting on α i1...jq . Therefore, for the purpose of computing the symbol, we note that ∂ and ∂ behave like on a complex manifold. The same reasoning works for ∂ * and ∂ * . Since∆ BC is elliptic on complex manifolds, this ends the proof.
The same considerations in the proof of Proposition 3.2 also prove that the ∂, ∂, and the Aeppli Laplacians are elliptic, too.
Denote by h p,q BC , h p,q A respectively the finite complex dimensions of H p,q BC and of H p,q A .

Bott-Chern harmonic (1, 1)-forms on almost Hermitian 4-manifolds
The goal of this section is to study the space of Bott-Chern harmonic forms of bidegree (1, 1) on almost Hermitian manifolds of real dimension 4. We start noting that this space is a conformal invariant of the metric. The two Hodge star operators behave, on the space A p,q , as * ω = e t(n−p−q) * ω .
Therefore, when p + q = n, the space H p,q BC is a conformal invariant of almost Hermitian metrics, thanks to the characterizatioñ In particular, h p,q BC is also a conformal invariant of almost Hermitian metrics for p + q = n. This is especially true when 2n = 4 and p = q = 1.
Let (M, g) be a compact oriented Riemannian manifold of real dimension 4, and set the subspace of harmonic anti-self-dual 2-forms and set b − = dim R H − . Note that b − is metric independent: see [6, Chapter 1] for its topological meaning.
BC is a conformal invariant of the metric, up to a conformal change of the Hermitian metric g, we can assume in this proof that ω is Gauduchon, i.e., ∂∂ω = 0.
We divide the proof in two steps.
(I) First, we prove that the space of Bott-Chern harmonic (1, 1)-forms is Let φ ∈ H 1,1 BC . By equation (7), we have φ = f ω + γ, where f is a smooth function with complex values on M and * γ = −γ. To prove the characterization of H 1,1 BC , we claim that f is a complex constant. Note that Expanding condition (10), using condition (9) and ∂∂ω = 0, we get We claim that the differential operator P ∶ C ∞ (M, C) → C ∞ (M, C) defined by is strongly elliptic, since its principal part is given by −i * (∂∂f ∧ ω).
Let us then verify that the differential operator L ∶ C ∞ (M, C) → C ∞ (M, C) defined by L ∶ f ↦ −i * (∂∂f ∧ ω) is strongly elliptic. Choose a local coframe {ζ 1 , ζ 2 } of bi-degree (1, 0) centered in a point m ∈ M and such that the almost Hermitian metric is written Let {V 1 , V 2 } be the corresponding dual frame. We have Wedging ∂∂f together with ω, we get where R is a differential operator which involves at most first order derivatives of f . Since Vol = ζ 1212 , it follows that which is strongly elliptic since −V 1 V 1 − V 2 V 2 is strongly elliptic, proving the claim.
Let us prove that the function f ∈ ker(P ) is constant. Note that P is a real differential operator, i.e., P (f ) = P (f ). Hence, f ∈ ker(P ) iff Re f ∈ ker(P ) and Im f ∈ ker(P ). By considering Re f and Im f instead of f , for the moment we may assume that f is real valued. Then, f ∶ M → R has a maximum and a minimum. Let m 0 ∈ M be a maximum point for f and set f (m 0 ) = N . Let U ∋ m 0 be a local chart and r > 0 such that B(m 0 , r) ⊂ U . The differential operator P is strongly elliptic on M , therefore it is strongly elliptic on B(m 0 , r). Since f ∈ ker(P ), by the maximum principle it follows that f is constant on B(m 0 , r). Since {m ∈ M ∶ f (m) = N } is both open and closed, f is constant on M .
In case (b), since for every element f ω + γ ∈ H 1,1 BC we have f = 0, it follows that H 1,1 BC coincides with the space of harmonic and anti-self-dual (1, 1)-forms H − C , yielding h 1,1 BC = b − . The theorem is proved.

Bott-Chern harmonic forms on the Kodaira-Thurston manifold
In this section we are going to compare Bott-Chern and Dolbeault harmonic forms on a family of almost Kähler structures on the Kodaira-Thurston manifold, following Holt and Zhang, [12].
The Kodaira-Thurston manifold, here denoted by M , is defined to be the direct product S 1 × (H 3 (Z) H 3 (R)), where H 3 (R) denotes the Heisenberg group for every t 0 , x 0 , y 0 , z 0 ∈ Z. The vector fields Consider the almost complex structure J b , for b ∈ R ∖ {0}, given by spanning T 1,0 p M at every point p ∈ M , along with their dual (1, 0)-forms Their structure equations are Endow every (M, J b ) with the family of almost Kähler metrics given by the compatible symplectic forms Define the volume form Vol such that Holt and Zhang, in [12], computed the spaces H p,q ∂ for every p, q. We will verify when H p,q BC = H p,q In [3,Section 6] it is proved that Bidegree By equation (12), note that V 2 (g) = 0, implying V 2 V 2 (g) = 0. The operator −V 2 V 2 is a real operator, and it is strongly elliptic when computed on functions depending only on the coordinates y, z. Consider the projection π ∶ M → T 2 = Z 2 R 2 given by π([t, x, y, z]) = ([t, x]). The fiber of π is a torus with coordinates y, z. As the fiber is compact, by the maximum principle applied to Re(g) and Im(g), we get that g is constant on each fiber. Therefore, the function g on M depends only on the coordinates t, x. Now, note that 4[V 1 , which derives from (13), and taking into account that g = g(t, x), we obtain V 2 V 2 (f ) = 0. Again by (12), note that V 1 (f ) = 0, implying V 1 V 1 (f ) = 0. Since [V 2 , V 2 ] = 0, we derive that f belongs to the kernel of −V 1 V 1 − V 2 V 2 , which is strongly elliptic. It follows that f is a complex constant.

Equation (12) also yields
Since f is a complex constant, it follows that g = 0. Therefore BC , see [3, Section 6] for the proof. It is also easy to see However, in [12], Holt and Zhang proved that σ = Ce 2πilx φ 2 ∈ H 0,1 ∂ , for l ∈ Z, b = 4πl and for any C ∈ C. Since σ ∉ H 0,1 BC , we just proved, for b = 4πl, Bidegree ( if and only if ∂∂ * s = 0 and ∂s = 0, i.e., iff It is an easy verification that the (2, 1)-form * σ does not satisfy the first equation of the system (14).
Remark 5.1. Counting the solutions (21), i.e., finding a lower bound on the complex dimension of H 1,2 BC , is equivalent to asking how many couples (l, m) ∈ Z 2 satisfy (20), which is equivalent to counting how many couples (l, m) ∈ Z 2 satisfy where we set d = b 8π. Counting the number of solutions can be thought of as asking how many lattice points in Z × Z lie on a circle with centre (d, 0) and radius d. This last number theoretic problem has already been addressed and solved by Holt and Zhang in [12,Section 4], where they show that by changing the choice of b (or equivalently d) one can make the number of solutions become arbitrarily large.
Therefore, in view of the argument as above, we infer that by changing our choice of b, h 1,2 BC may become arbitrarily large. This conclusion has been already obtained by Holt in [11,Example 4.4] (the case ρ = 1 in the notation of Holt), where the space of Bott-Chern harmonic (1, 2)-forms is fully characterized.
Summarizing the results just obtained, we state the following proposition.
Their structure equations are Endow (M, J) with the almost Hermitian metric given by the compatible symplectic form and define the volume form Vol such that Note that b − = 1. Also note that ω is strictly locally conformally almost Kähler, since dω = πe 134 = θ ∧ ω, with θ = πe 4 , which is closed but not exact by (22).
If x 2 = 0, it is easy to see that (k, l) ≠ (0, 0) implies A k,l,m,n = B k,l,m,n = C k,l,m,n = 0. Therefore, the functions A, B, C depend only on variables x 2 , y 2 , and we can assume k = l = 0. For every (0, 0) ≠ (m, n) ∈ Z 2 , system (24) rewrites into for every z 1 = x 1 + iy 1 , z 2 = x 2 + iy 2 ∈ C and 0 ≠ K ∈ C, thus the functions A, B, C in (27) are not well defined on M . Other solutions of system (23) are found when k = l = m = n = 0 and A, B, C ∈ C are complex constants. More precisely, we get 2A = if and B = −C.
In general, the spaces H p,q BC and H p,q A seem to be unrelated to the Bott-Chern and Aeppli cohomology spaces just introduced.
However, if we take n = dim R M = 4 and p = q = 1, note that A 1,1 s = A 1,1 = A 1,1 r , and the previous Bott-Chern decomposition yields, intersecting with ker d, ker d ∩ A 1,1 = H 1,1 BC ⊕ ker d ∩ ∂∂A 0,0 . Therefore, there is a well defined injection In general, there seems no reason to think this injection is also a surjection. Note that j being surjective would imply h 1,1 BC is an almost complex invariant on 4manifolds.
We can re-obtain the previous injection of (1, 1)-forms as a particular case of the following observation. Let (M, J, g, ω) be a compact almost Hermitian manifold of real dimension 2n and intersect the Bott-Chern decomposition with the space ker d ∩ A p,q s , deriving ker d ∩ A p,q s = H p,q BC ∩ A p,q s ⊕ ker d ∩ A p,q s ∩ ∂∂A p−1,q−1 . Therefore, there is a well defined injection