On non-pure forms on almost complex manifolds

T.-J. Li and W. Zhang defined an almost complex structure $J$ on a manifold $X$ to be {\em \Cpf}, if the second de Rham cohomology group can be decomposed as a direct sum of the subgroups whose elements are cohomology classes admitting $J$-invariant and $J$-anti-invariant representatives. It turns out (see T. Draghici, T.-J. Li and W. Zhang) that any almost complex structure on a 4-dimensional compact manifold is \Cpf. We study the $J$-invariant and $J$-anti-invariant cohomology subgroups on almost complex manifolds, possibly non compact. In particular, we prove an analytic continuation result for anti-invariant forms on almost complex manifolds.


Introduction
Let (X, J, ω) be a compact Kähler manifold of complex dimension n. Then the celebrated Hodge decomposition Theorem states the (p, q)-decomposition on forms transfers at the cohomology level, namely, H k (X; C) ∼ = p+q=k H p,q ∂ (X), where H p,q ∂ (X) denotes the (p, q)-Dolbeault group. In order to generalize the previous decomposition to arbitrary compact almost complex manifolds, in [16] T.-J. Li and W. Zhang introduced the notion of C ∞ -pure-and-full almost complex structures. An almost complex structure J on a 2n-dimensional manifold X is said to be C ∞ -pure-and-full if H 2 (X; R) = H (X) R ⊂ H 2 (M ; R) can be thought as a generalization of real Dolbeault group H 2,0 ∂ (X) ⊕ H 0,2 ∂ (X) ∩ H 2 (X; R) to the non-integrable case. For other results on C ∞ -pure-and-full almost complex structures see e.g. [10], [1], [3].
In dimension higher than four, there are examples of non C ∞ -pure and non C ∞ -full almost complex structures on compact manifolds. Nevertheless, in [9] and [12], it is proved that if (X, ω) is a compact symplectic manifold of dimension 2n, then every ω-compatible almost complex structure J is C ∞ -pure.
In this paper we are interested in studying the J-invariant and J-anti-invariant respectively cohomology subgroups on almost complex manifolds X 2n , possibly non compact. In particular we will search for pairs (ω, J) where ω is either a J-invariant or J-anti-invariant closed 2-form. We are also interested in determining whether ω can also be taken to be symplectic. We note that symplectic forms always have compatible almost complex structures, which are invariant in our language. Therefore in the symplectic case we will focus on finding anti-invariant almost complex structures.
After some preliminaries in section 1, in section 2 we start with the observation that if n is odd and (ω, J) is an anti-invariant pair then ω n is identically zero, see Proposition 2.1. In particular ω cannot be symplectic. However it is not true that lower powers of ω must vanish, even in cohomology. In Example 2.2 we provide an example of a 2-cohomology class on a 6-dimensional compact nilmanifold N endowed with a left-invariant almost complex structure J, which belongs to H (N ) R and whose square is non-zero. In searching for anti-invariant forms with ω symplectic then we must assume that n is even. We consider the case of open manifolds. Theorem 3.1 says that on an open symplectic 4 manifold (X 4 , ω) there exists an anti-invariant J if and only if the tangent bundle is symplectically trivial. However Proposition 3.3 shows that this is no longer true in higher dimensions. If we do not fix the symplectic form, Theorem 3.5 says that given any open manifold X 2n with n even and T X trivial, and any cohomology class a ∈ H 2 (X, R), there exists an anti-invariant pair (ω, J) with [ω] = a.
In the remainder of the paper we are motivated by the problem of finding anti-invariant pairs (ω, J) in a given cohomology class without the restrictive symplectic assumption. As in Proposition 2.1 it is not hard to show, for example, that on a 6-manifold such an ω must everywhere have rank 0 or 4. There are no other pointwise restrictions. Still, this seems restrictive, but we show in section 5 that for many codimension 2 submanifolds W we can find closed forms ω supported in arbitrarily small tubular neighborhoods such that ω has everywhere rank 0 or 4 and [ω] is Poincaré dual to W . Given this, it is natural to ask whether anti-invariant pairs can be constructed using cut and paste methods.
Section 4 shows that this is impossible, indeed, despite the rank condition, the compactly supported forms constructed in section 5 have no anti-invariant almost complex structures. This is a consequence of the unique continuation theorem, see Theorem 4.1. This says that given an anti-invariant pair (ω, J), if ω vanishes on an open subset then it must vanish everywhere on a connected X 2n .
1. Preliminaries 1.1. C ∞ -pure-and-full almost complex structures. Let X be a 2n-dimensional manifold (without boundary) and J be a smooth almost complex structure on X. Let (T 1,0 X) * and (T 0,1 X) * be the bundle of complex 1-forms on X of type (1, 0) and (0, 1), respectively. Denote by Λ k C (X) the bundle of C ∞ complex k-forms on X and by the same symbol Λ k C (X) = Γ(M, Λ k C (X)) the space of sections of Λ k C (X). Then, as usual, setting Λ p,q J (X) = Λ p (T 1,0 X) * ∧ Λ q (T 0,1 X) * , the space of complex k-forms decomposes as Λ k C (X) = (X) R and we will refer to them as the spaces of J-invariant and J-anti-invariant forms, respectively. For a finite set S of pairs of integers, let We recall the following (see [

1.2.
Pure-and-full almost complex structures. Let (X, J) be a compact 2ndimensional almost complex manifold. Denote by D k (X) the space of currents of dimension k (or degree (2n − k)), i.e., the topological dual of the space Λ k (X) of r-forms on X (see e.g [8], [7]). The exterior differential d on Λ • (X) induces a differential on D • (X), still denoted by d. Then the de Rham homology H • (X; R) is the cohomology of the differential complex (D • (X), d) and H k dR (X; R) ≃ H 2n−k (X; R). We will denote by P : H 2n−k (X; R) → H k dR (X; R) this isomorphism. As in the case of complex forms, the almost complex structure J induces a bi-grading on the space D C k (X) of complex currents of dimension k. Hence, the spaces of currents given by the homology classes represented by a real current of bidimension (2, 0) + (0, 2) (respectively, (1, 1)). Then we have the following definition due to T.-J. Li and W. Zhang. ). An almost complex structure J on X is said to be: • pure-and-full if it is both pure and full, i.e. if the following decomposition holds: The notions of C ∞ -pure/full and pure/full almost complex structures can be given in a similar way for k-forms and k-currents, respectively.
The relations between being C ∞ -pure-and-full and being pure-and-full are summarized in the following.
In the sequel we will use the following notation:

be the product of the special unitary group with the torus
Define the almost complex structure J on SU (2) × T 3 by giving the following complex (1, 0)-forms then (see [2, example 5.4]) we have that J is a C ∞ -full non C ∞ -pure almost complex structure. Hence, accordingly to Proposition 1.3, J is a pure non full almost complex structure on SU (2) × T 3 .

Compact manifolds
2.1. J-Invariant cohomology classes. For general 2-cohomology classes on arbitrary manifolds, we can show ; furthermore, there exists a point p ∈ X such that ϕ n (p) = 0. Since ϕ ∈ Λ − J (X) and n is odd, we have that Jϕ n = −ϕ n . Arguing as in the proof of i), we obtain that α n = 0. This is absurd.
As a direct consequence of the last Proposition, in the compact case we have the following Corollary 2.2. Let (X, ω) be a 2n-dimensional compact symplectic manifold, with n odd. Let J be an almost complex structure on M . Then [ω] / ∈ H − J (X). The following provides an example of a 2-cohomology class a on a 6-dimensional compact nilmanifold N endowed with a left-invariant almost complex structure J, which belongs to H + J (N ) ∩ H − J (N ) and whose square is non-zero; nevertheless, a 3 = 0, according to Proposition 2.1.

2.2.
Example. Let g be the 6-dimensional nilpotent Lie algebra whose dual vector space admits a basis {e 1 , . . . , e 6 } satisfying the following Maurer-Cartan equations: where e ij = e i ∧ e j and so on. Let G be the connected and simply-connected Lie group having g as Lie algebra. Then, G admits a uniform discrete subgroup Γ and N = Γ\G is a 6-dimensional nilmanifold. Let J be the almost complex structure on N defined by the following global (1, 0)-forms ψ 1 = e 1 + ie 2 , ψ 2 = e 4 + ie 6 , ψ 3 = e 3 + ie 5 .  Proof. Let {V 1 , . . . , V n , W 1 , . . . , W n } be a global frame on X, J 0 be the almost complex structure on X defined as J 0 V i = W i , i = 1, . . . , n, and g be the Hermitian metric defined by requiring that {V 1 , . . . , V n , W 1 , . . . , W n } is an orthonormal frame. Then the fundamental form ω 0 of g is an almost symplectic form with symplectically trivial tangent bundle. Denote by Then S a symp ⊂ S symp ⊂ S symp and according to the h-principle (see [13] or [11, 10.2.2]) the inclusion S a symp ֒→ S symp is an homotopy equivalence. In other words there exists a family ω t of almost symplectic forms on X with ω 1 symplectic and a = [ω 1 ].

By
Denote by E → X × [0, 1] the bundle over X × [0, 1] whose fibre is given by Since (X, ω 0 ) is symplectically trivial, then there exists a section of E over X × {0}. By the homotopy lifting property there is a section of E over X × [0, 1]. In particular (X, ω 1 ) is symplectically trivial. Hence, in view of Remark 3.2, it follows that there exists an almost complex structure J such that ω 1 is a J-anti-invariant form.

Analytic continuation of anti-invariant forms
In this section we prove an analytic continuation result for anti-invariant closed forms on almost complex manifolds. The proof in the 4-dimensional case is a direct consequence of Hodge theory.
The general case is more subtle but eventually reduces again to Aronszajn's Theorem.

Proof. (general case)
Let D be a J-holomorphic disk in X. We can identify a small neighborhood of D ⊂ X with the total space of a complex vector bundle D×C n−1 → D such that J| D×C n−1 coincides with the standard product structure along D × {0}. Let t : (z, w) → (z, tw) denote the scaling in the fibers.
We linearize J on D × C n−1 by replacing J by Then J 0 is an almost complex structure on D×C n−1 invariant under scaling and restricting to the standard complex structure on the fibres {x} × C n−1 . We can also replace α by α 0 defined by Claim. α 0 is a closed, smooth J 0 -anti-invariant form on D × C n−1 . It vanishes on the fibers {x} × C n−1 .
To justify the claim we compute the almost complex structures and anti-invariant forms explicitly in local coordinates.
Let (x 1 , . . . , x 2n ) be local coordinates in a neighborhood of a point 0 ∈ D × {0} such that the scaling is given by t(x 1 , . . . , x 2n ) = (x 1 , x 2 , tx 3 , . . . , tx 2n ). Denote the coordinate vector fields by e i = ∂ ∂x i . Then t * e i = e i for i = 1, 2 and t * e i = te i for i > 2. We can write α = i<j a ij dx i ∧ dx j and where the a ij and b ij are functions of (x 1 , . . . , x 2n ). Now, as the zero section is holomorphic and the fibers of the normal bundle are holomorphic along the zero section we may assume that b 1j (0) = δ 2j , b 2j (0) = −δ 1j and b i1 (0) = b i2 (0) = 0 for all i > 2. Furthermore, we will also suppose that coordinates are chosen such that the matrix (b ij ) i,j>2 is the standard complex structure at all points (x 1 , x 2 , 0 . . . , 0). Next, as anti-holomorphic forms necessarily vanish on complex planes we have that a 12 (0) = 0. We compute at a fixed point x in the fiber over 0.
Similarly J t (e 2 (x)) is a smooth vector field converging to as t → 0. Hence in these coordinates the fibers converge to complex vector spaces as t → 0. For the forms, we have α t (x)(e 1 , e 2 ) = 1 t α(tx)(e 1 , e 2 ) = 1 t a 12 (tx) (0) as t → 0. As the convergence here is smooth, α 0 is closed and J 0 -anti-invariant as claimed. Now, by the computations above, to understand α 0 we are only interested in the values of a 1i and a 2i for i > 2 along the zero-section, and hence will think of these only as functions of x 1 and x 2 . The function a 12 on the other hand we must still consider as a function of x 1 , . . . , x 2n , although to simplify notation it is enough to consider the case when it is linear when restricted to the fibres.
There are relations amongst the functions above which we now detail. First of all, as a limit of closed forms, α 0 is closed. Calculating its dx 1 ∧ dx 2 ∧ dx i component for i > 2 we get Now we consider the anti-invariance of α 0 at points along the zero-section and obtain (2) a 1,(2i−1) = −a 2,2i , a 1,2i = a 2,(2i−1) for i > 1. For instance, to prove the first of these, we use the formula α 0 (0)(e 1 , e 2i−1 ) = −α 0 (0)(e 2 , e 2i ). Next, using our formulas for α 0 , we compute 0 = α 0 (e 1 , J 0 e 1 ) = α 0 (e 1 , e 2 + j,k>2 As this holds for all choices of x k we obtain Applying equations (2), (1) and then (3) we obtain for all i > 1, and similarly Differentiating (4) with respect to x 2 gives and differentiating (5) with respect to x 1 gives Adding equations (6) and (7) gives a formula for ∆a 1,(2i−1) in terms of the a 1j for j > 2 and their first derivatives. In fact, if we think of the a 1j , a 2k as giving a function a : D → R 4(n−1) , (x 1 , x 2 ) → (a 13 , . . . , a 1,2n , a 23 , . . . , a 2,2n ), then the above calculations give a bound on the Laplacian ∆a in terms of bounds on the function and its first derivatives ∂a ∂x 1 and ∂a ∂x 2 . Thus by Aronszajn's Theorem the function a satisfies a unique continuation theorem, in particular if a is identically zero near a point then it is zero everywhere. Hence if the function a vanishes near a point in D then it vanishes everywhere on D. As a 12 is identically zero along the zero-section D (as anti-invariant forms vanish on holomorphic curves) we conclude that if α = 0 near a point of D then it is zero at all points of D.
To conclude the proof we need the following lemma.
Lemma 4.2. Fix a Riemannian metric on X defining a distance function d and let K ⊂ X be compact. There exists an ǫ > 0 such that for any x, y ∈ K with d(x, y) < ǫ there exists a holomorphic disk D in X passing through x and y.
This follows from [17], Theorem 3.1.1 (i). The lemma, together with the unique continuation theorem established along disks, implies that the set of points where α vanishes identically in a neighborhood is open. As it is clearly closed we can conclude the proof of Theorem 4.1.

Closed forms of rank 0 and 4
In this section we construct closed forms which have everywhere rank 0 or 4, have compact support near a submanifold W 2n−2 of X 2n , and are Poincaré dual to [W ] ∈ H 2 (X). As explained in the introduction, although there are no pointwise obstructions, the unique continuation theorem, Theorem 4.1, shows that there do not exist corresponding anti-invariant almost complex structures. This is our main proposition.
Proposition 5.1. Let W ⊂ X 2n be a (2n − 2)-dimensional compact submanifold with trivial normal bundle ν(W ) and σ be a 1-form on W such that dσ never vanishes and d(σ ∧ dσ) = 0. Then there is a compactly supported 2-form ω on the total space of ν(W ) which everywhere has rank either 4 or 0 and is cohomologous to the Thom class τ of ν(W ).
Proof. Let τ = r(x 1 , y 1 )dx 1 ∧dy 1 , be the local expression of the Thom class of ν(W ), where r(x 1 , y 1 ) is a bump function such that r = 1 on a neighborhood of W in T W . Define ω = τ + d(rσ) .