On Symplectically Harmonic Forms on Six-dimensional Nilmanifolds

R. IBÁÑEZ ^{1}, YU. RUDYAK ^{2}, A. TRALLE ^{3}, L. UGARTE ^{4}

1Departamento de Matemáticas, Facultad de Ciencias, Universidad del Pais Vasco, Apdo. 644, 48080 Bilbao, Spain, E-mail: mtpibtor@lg.ehu.es

2 FB6/ Mathematik, Universität Siegen, 57068 Siegen, Germany, E-mail: rudyak@mathematik.uni-siegen.de

3Department of Mathematics, University of Warmia and Mazura, 10561 Olsztyn, Poland, E-mail: tralle@tufi.wsp.olsztyn.pl

4Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain, E-mail: ugarte@posta.unizar.es

Abstract

In the present paper we study the variation of the dimensions h_{k} of spaces of symplectically harmonic cohomology classes (in the sense of Brylinski) on closed symplectic manifolds. We give a description of such variation for all 6-dimensional nilmanifolds equipped with symplectic forms. In particular, it turns out that certain 6-dimensional nilmanifolds possess families of homogeneous symplectic forms ω_{t} for which numbers h_{k}(M, ω_{t}) vary with respect to t. This gives an affirmative answer to a question raised by Boris Khesin and Dusa McDuff. Our result is in contrast with the case of 4-dimensional nilmanifolds which do not admit such variations by a remark of Dong Yan.

Mathematics Subject Classification (2000): 53D05.

PACS numbers: 02.40.Ma, 03.20.+i, 0.3.65.-w Key words and phrases: symplectically harmonic form, Brylinski’s conjecture, Lefschetz map

1 Introduction

Given a symplectic manifold (M^{2m}, ω), we denote by [ω] ∈ H^{2}(M) the de Rham cohomology class of ω. Here the notation M^{2m} means that M is a 2m-dimensional manifold. Furthermore, we denote by L_{ω} : Ω^{k}(M) → Ω^{k+2}(M) the multiplication by ω and by L_{[ω]} : H^{k}(M) → H^{k+2}(M) the induced homomorphism in the de Rham cohomology H^{∗}(M) of M. As usual, we write L instead of L_{ω} or L_{[ω]} if there is no danger of confusion. We say that a symplectic manifold (M^{2m}, ω) satisfies the Hard Lefschetz condition if, for every k, the homomorphism

L^{k} : H^{m−k}(M) → H^{m+k}(M)

is surjective. In view of the Poincaré duality, for closed manifolds M it means that every L^{k} is an isomorphism.

In 1988 J. L. Brylinski [2] introduced the concept of symplectically harmonic forms (resp. Poisson harmonic forms), defined for any symplectic manifold (resp. Poisson manifold). Further he conjectured that on compact symplectic manifolds, every de Rham cohomology class has a symplectically harmonic representative. In fact, this conjecture asks about the possibility of constructing of a symplectic Hodge theory.

Brylinski proved that this conjecture is true for compact Kähler manifolds. However, it is not true in general, as it was shown in [5, 9, 15]. Mathieu [9] proved the following theorem, which applies to an arbitrary (not necessarily compact) symplectic manifold.

Theorem 1.1 A symplectic manifold (M^{2m}, ω) satisfies the Brylinski conjecture if and only if it satisfies the Hard Lefschetz condition. In other words, the following assertions are equivalent:

(i) every de Rham cohomology class has a symplectically harmonic representative;

(ii) for every k ≤ m, the homomorphism L^{k} : H^{m−k}(M) ⟶ H^{m+k}(M) is surjective.

Mathieu’s proof involves the representation theory of quivers and Lie superalgebras. An alternative and nice proof can be found in the paper of Yan [15], who studies a special type of infinite dimensional 𝔰𝔩(2)-representation and, basing on this, proves a duality theorem for symplectically harmonic forms which, in turn, implies Theorem 1.1.

Given a symplectic manifold (M^{2m}, ω), let Ω_{hr}^{k}(M) = Ω_{hr}^{k}(M, ω) denote the subspace of symplectically harmonic forms of Ω^{k}(M). The form α is called symplectically harmonic if dα = 0 = δα, where δ = (−1)^{k+1} ∗ d∗ and ∗ is the symplectic star operator, see Section 2. We set

H_{hr}^{k}(M) = H_{hr}^{k}(M, ω) := Ω_{hr}^{k}(M)/(Im d ∩ Ω_{hr}^{k}(M))

and

h_{k}(M) = h_{k}(M, ω) := dim H_{hr}^{k}(M, ω).

Since every symplectically harmonic form is closed, H_{hr}^{k}(M) is a subgroup of H^{k}(M) and h_{k} ≤ b_{k}. Of course, the above definition of the numbers h_{k} is not symmetric: one can also consider the “dual” numbers

h_{k}^{∗}(M) = h_{k}^{∗}(M, ω) := dim (Ω_{hr}^{k}/Im δ ∩ Ω_{hr}^{k}) .

It turns out that the duality isomorphism ∗ : Ω_{hr}^{m−k} → Ω_{hr}^{m+k} yields the equality h_{m−k} = h_{m+k}^{∗}, see Section 7. Relating with the study of symplectically harmonic forms, we are interested in the following question raised by Boris Khesin and Dusa McDuff, see Yan [15]. Question: Which compact manifolds M possess a continuous family ω_{t} of symplectic forms such that h_{k}(M, ω_{t}) varies with respect to t? This question, according to Khesin, is probably related to group theoretical hydrodynamics and geometry of diffeomorphism groups. Some indirect indications for an existence of such relations can be found in [1]. Since we want to consider manifolds as in the question, it makes sense to give them a certain name. So, let us call a closed smooth manifold M flexible if M possesses a continuous family of symplectic forms ω_{t}, t ∈ [a, b], such that h_{k}(M, ω_{a}) ≠ h_{k}(M, ω_{b}) for some k. So, the above question asks about the existence of flexible manifolds. Yan [15] has studied the case of closed 4-manifolds. He proved that 4-dimensional nilmanifolds are not flexible. He has also found examples of flexible 4-manifolds. Actually, one step in this proof of the existence is wrong, but the whole proof can easily be repaired, and hence the existence result holds. See Section 4 for details. So, passing to higher dimensions, we have the following question: Question: Do there exist flexible nilmanifolds of dimension ≥ 6? Related with this question, we show that some 6-dimensional nilmanifolds are flexible. Some words about tools. First, in order to prove the flexibility, we must be able to compute the symplectically harmonic Betti numbers. It turns out that, for every closed symplectic manifold (M^{2m}, ω) and k = 1 or 2 we have

h_{2m−k}(M, ω) = rank (L_{[ω]}^{m−k} : H^{k}(M) → H^{2m−k}(M)),

see 3.2 and 3.4. So, a purely cohomological information is enough in order to compute h_{2m−k}. Furthermore, if h_{2m−k}(M, ω_{1}) ≠ h_{2m−k}(M, ω_{2}), k = 1, 2, then M is flexible, see 2.11 and 3.2. In general, not only symplectically harmonic forms but also Poisson harmonic forms are being of great interest in different areas of mathematics and physics, [3]. We use the sign QED in order to indicate the end of a proof. However, if we formulate a claim without proof, we put the sign □ in the end of the claim.

2 Symplectically harmonic forms

Let (M^{2m}, ω) be a symplectic manifold. It is well known that there exists a unique nondegenerate Poisson structure Π associated with the symplectic structure (see, for example, [8, 14]), that is, Π is a skew-symmetric tensor field of order 2 such that [Π, Π] = 0, where [−, −] is the Schouten-Nijenhuis bracket.

The Koszul differential δ : Ω^{k}(M) ⟶ Ω^{k−1}(M) is defined for symplectic manifolds, and more generally, for Poisson manifolds as

δ = [i(Π), d].

Brylinski have proved in [2] that the Koszul differential is a symplectic codifferential of the exterior differential, with respect to the symplectic star operator. We choose the volume form associated to the symplectic form, that is, v_{M} = ω^{m}/m!. Then we define the symplectic star operator

∗ : Ω^{k}(M) ⟶ Ω^{2m−k}(M),

by the condition β ∧ (∗α) = Λ^{k}(Π)(β, α)v_{M}, for all α, β ∈ Ω^{k}(M). The symplectic star operator satisfies the identities

∗^{2} = id, δ = (−1)^{k+1} ∗ d∗, and i(Π) = L^{∗} := − ∗ L ∗ .

Furthermore, if M is a symplectic manifold M then the operators L, d, δ and L^{∗} (acting on the algebra Ω^{∗}(M)) satisfy the following commutator relations:

[L, d] = 0, [L, δ] = −d, [L^{∗}, δ] = 0, [L^{∗}, d] = −δ.  (2.1)

Remark 2.1 The symplectic star operator was first considered by Libermann [8] as ∗(α) = i(μ^{−1}α)v_{M}, where μ is the canonical isomorphism between the exterior algebras of vector fields and forms. She has also introduced and studied the operators δ = (−1)^{k+1}∗d∗ and L^{∗} = − ∗ L∗. In particular, she has proved (using the Lepage decomposition) that

{α ∈ Ω^{m−k}(M) | L^{k+1}α = 0} = {α ∈ Ω^{m−k}(M) | L^{∗}α = 0}.  (2.2)

(See [15] for a proof using the theory of 𝔰𝔩(2)-representations.)

Definition 2.2 A k-form α on the symplectic manifold M is called symplectically harmonic if dα = δα = 0. We denote the space of harmonic k-forms by Ω_{hr}^{k}(M).

It is clear from (2.1) that the form ω ∧α is symplectically harmonic whenever α is. Hence, for every k we have the mapping L : Ω_{hr}^{k}(M) ⟶ Ω_{hr}^{k+2}(M).

We set

H_{hr}^{k}(M) = Ω_{hr}^{k}(M)/Im d ∩ Ω_{hr}^{k}(M) and h_{k} = h_{k}(M, ω) = dim H_{hr}^{k}(M).

So, for every symplectic manifold M, its de Rham cohomology H^{∗}(M) contains a symplectically harmonic subspace H_{hr}^{∗}(M). We say that a de Rham cohomology class is symplectically harmonic if it contains a symplectically harmonic representative, i.e. if it belongs to the image of the inclusion H_{hr}^{∗}(M) ⊂ H^{∗}(M). Finally, we say that a manifold M is flexible if M possesses a continuous family of symplectic forms ω_{t}, t ∈ [a, b], such that h_{k}(M, ω_{a}) ≠ h_{k}(M, ω_{b}) for some k.

Remark 2.3 In the general case of a (degenerate) Poisson manifold (M, Π), we say that a k-form α is Poisson harmonic if dα = 0 = δα. Notice that for a Poisson manifold, in particular for a symplectic manifold, Δ = dδ + δd ≡ 0, contrarily to the Riemannian case.

Remark 2.4 Recall that a symplectomorphism between two symplectic manifolds (M, ω_{1}) and (N, ω_{2}) is a diffeomorphism ϕ : M ⟶ N such that ϕ^{#}(ω_{2}) = ω_{1}. It is easy to see that

ϕ^{∗}(H_{hr}^{k}(N)) = H_{hr}^{k}(M).

where ϕ^{∗} : H^{k}(N) → H^{k}(M) is the induced homomorphism in the de Rham cohomology. In other words, H_{hr}^{∗}(−) is a symplectic invariant. In particular, if h_{k}(M, ω_{1}) ≠ h_{k}(M, ω_{2}) for two symplectic forms ω_{1}, ω_{2} on M then ω_{1} and ω_{2} are not symplectomorphic.

We do not know whether a smooth map (not a diffeomorphism) ψ with ψ^{#}ω_{2} = ω_{1} induces a map of symplectically harmonic cohomology.

Proposition 2.5 ([15]) For every symplectic manifold (M, ω), the homomorphism

L^{k} : Ω_{hr}^{m−k}(M) ⟶ Ω_{hr}^{m+k}(M)

is an isomorphism. □

Corollary 2.6 The homomorphism

L^{k} : H_{hr}^{m−k}(M) ⟶ H_{hr}^{m+k}(M)

is an epimorphism. In particular, h_{m−k} ≥ h_{m+k}. □

Corollary 2.7 Let (M^{2m}, ω) be a symplectic manifold. Then

H_{hr}^{m+k}(M) = Im{L^{k} : H_{hr}^{m−k}(M) → H^{m+k}(M)} ⊂ H^{m+k}(M).

Proof : It is a direct consequence of the commutativity of the diagram

Ω_{hr}^{m−k}(M) ⟶^{L^{k}} Ω_{hr}^{m+k}(M)
↓  ↓
H_{hr}^{m−k}(M) ⟶^{L^{k}} H_{hr}^{m+k}(M)

(2.3)

since the top map L^{k} is an isomorphism by 2.5 and both vertical maps are the epimorphisms.

QED

Corollary 2.8 Let (M^{2m}, ω) be a closed symplectic manifold. If h_{m+k}(M) = b_{m+k}(M) then h_{m−k}(M) = b_{m−k}(M). □

Proof : Because of 2.6 and Poincaré duality, h_{m−k} ≥ h_{m+k} = b_{m+k} = b_{m−k} ≥ h_{m−k}.

QED

Corollary 2.9 Let (M, ω) be a symplectic manifold. If b_{k}(M) = 0 for some k ≤ m, then h_{2m−i}(M, ω) = 0 for i ≤ k with k − i even.

Proof : It follows from 2.7, since the homomorphism

L^{m−i} : H^{i}(M) ⟶ H^{2m−i}(M)
passes through the trivial group H^{k}(M).  QED

Lemma 2.10 Let 𝕃 be the space of all linear maps ℝ^{k} → ℝ^{l}. Then the following holds:

(i) for every r the set

{A ∈ 𝕃 | rank A ≤ r}

is an algebraic subset of 𝕃. (Here we regard 𝕃 as the space ℝ^{kl} of l × k-matrices whose entries are regarded as the coordinates);

(ii) for every m ≤ min{k, l} the set {A ∈ 𝕃 | rank A ≥ m} is open and dense in 𝕃;

(iii) let A, B ∈ 𝕃 be two linear maps such that rank A < rank B. Then the set

Λ = {λ ∈ ℝ | rank (A + λB) ≥ rank B}

is an open and dense subset of ℝ.

Proof : (i) This claim follows, because the rank of a matrix is equal to the order of the largest non-zero minor.

(ii) This claim follows from (i).

(iii) By (i), the set ℝ \ Λ is an algebraic subset of ℝ. So, it suffices to prove that Λ ≠ ∅.

But, by (ii), rank (B + μA) > rank A for μ small enough, and so Λ ≠ ∅.  QED

Corollary 2.11 Let ω_{0} and ω_{1} be two symplectic forms on a manifold M^{2m}. Suppose that, for some k > 0, h_{m−k}(ω_{0}) = h_{m−k}(ω_{1}), but h_{m+k}(ω_{0}) < h_{m+k}(ω_{1}). Then, for every ε > 0, there exists λ ∈ (0, ε) such that h_{m+k}(ω_{0} + λω_{1}) > h_{m+k}(ω_{0}). Moreover, M is flexible provided that it is closed.

Proof : The existence of λ follows from 2.7 and 2.10(iii). The flexibility of M follows, since ω_{0} + tω_{1} is a symplectic form for t small enough. Indeed, if we set ω_{t} = ω_{0} + tω_{1}, t ∈ [0, λ],

then h_{m+k}(ω_{0}) < h_{m+k}(ω_{λ}).  QED

Corollary 2.12 Let (M^{2m}, ω_{0}) be a closed symplectic manifold. Given k with 0 < k < m, suppose that h_{m−k}(M, ω) = b_{m−k}(M) for every symplectic form ω on M. Furthermore, suppose that there exists x ∈ H^{2}(M) such that

rank {L_{x}^{k} : H^{m−k}(M) → H^{m+k}(M)} > h_{m+k}(M, ω_{0}).

Then M is flexible.

Proof : Take a closed 2-form α which represents x. Then ω_{0} + tα is a symplectic form for t small enough. Furthermore, by 2.7 and 2.10(iii), there exists arbitrary small λ such

that h_{m+k}(ω_{0} + λα) > h_{m+k}(ω_{0}). Now the result follows from 2.11.  QED

We set

Ω_{sympl}(M) = {ω ∈ Ω^{2}(M) | ω is a symplectic form on M}

and define Ω(b, k) = {ω ∈ Ω_{sympl} | h_{k}(M, ω) = b}.

Corollary 2.13 Let M^{2m} be a manifold that admits a symplectic structure. Suppose that, for some k > 0, h_{m−k}(M, ω) does not depend on the symplectic structure ω on M. Then the following three conditions are equivalent:

(i) the set Ω(b, m + k) is open and dense in Ω_{sympl}(M);

(ii) the interior of the set Ω(b, m + k) in Ω_{sympl}(M) is non-empty;

(iii) the set Ω(b, m + k) is non-empty and h_{m+k}(M, ω) ≤ b for every ω ∈ Ω_{sympl}(M).

Proof : (i) ⇒ (ii). Trivial.

(ii) ⇒ (iii). Suppose that there exists ω_{0} with h_{m+k}(M, ω_{0}) > b. Take ω in the interior of Ω(b, m + k). Then, in view of 2.10, there exists an arbitrary small λ such that h_{m+k}(ω + λω_{0}) > b, i.e. ω does not belong to the interior of Ω(b, m + k). This is a contradiction.

(iii) ⇒ (i). This is true because of 2.10(ii).  QED

So, the family {Ω(b, m + k)|b = 0, 1, …} gives us a stratification of Ω_{sympl}(M) where the maximal strat is open and dense.

Lemma 2.14 Let (M^{2m}, ω) be a closed symplectic manifold, and set

ρ_{2k+1} = rank \{L^{m−2k−1} : H^{2k+1}(M) ⟶ H^{2m−2k−1}(M)\} , k = 0, …, [\frac{m − 1}{2}] .

Then ρ_{2k+1} is an even number. Furthermore, h_{2m−2k−1} ≤ ρ_{2k+1} ≤ b_{2k+1}, and ρ_{2k+1} = b_{2k+1} if and only if L^{m−2k−1} : H^{2k+1}(M) ⟶ H^{2m−2k−1}(M) is surjective.

Proof : Let p : H^{2k+1}(M) ⊗ H^{2m−2k−1}(M) ⟶ ℝ be the usual non-singular pairing given by

p ([α], [γ]) = ∫_{M} α ∧ γ,

for [α] ∈ H^{2k+1}(M) and [γ] ∈ H^{2m−2k−1}(M). Define a skew-symmetric bilinear form ⟨−, −⟩ : H^{2k+1}(M) ⊗ H^{2k+1}(M) ⟶ ℝ via the formula

⟨[α], [β]⟩ = p ([α], L^{m−2k−1}[β]) ,

for [α], [β] ∈ H^{2k+1}(M). It is easy to see that the rank of ⟨−, −⟩, which must be an even number 2l with 0 ≤ 2l ≤ b_{2k+1}, is equal to ρ_{2k+1}.

The inequality h_{2m−2k−1} ≤ ρ_{2k+1} follows from 2.7.

The last claim holds since, by the Poincaré duality, b_{2k+1}(M) = b_{2m−2k−1}(M).  QED

Remark 2.15 Lemma 2.14 yields the following well-known fact: if M in 2.14 satisfies the Hard Lefschetz condition, then all odd-dimensional Betti numbers are even.

Corollary 2.16 Let (M^{2m}, ω) be a closed symplectic manifold. If b_{2k+1}(M) is odd then h_{2m−2k−1} < b_{2m−2k−1} = b_{2k+1}. In particular, if b_{2k+1} = 1 then h_{2m−2k−1} = 0. □

Corollary 2.17 If b_{4k+2}(M) = 1 then ρ_{2k+1} does not depend on symplectic structure on M.

Proof : Because of the Poincaré duality, b_{2m−4k−2} = 1. So, [ω]^{m−2k−1}, and hence

L^{m−2k−1} : H^{2k+1}(M) ⟶ H^{2m−2k−1}(M),
is determined uniquely up to non-zero multiplicative constant.  QED

3 The numbers h_{k} and h_{2m−k} for k small

Proposition 3.1 Let (M, ω) be a symplectic manifold, and let k be a non-negative integer number such that the following holds:

(i) L^{k+2} : H^{m−k−2}(M) ⟶ H^{m+k+2}(M) is surjective;

(ii) If a ∈ H^{m−k−2}(M) is not symplectically harmonic, then La = 0.

Then every cohomology class in H^{m−k}(M) is symplectically harmonic.

Proof : Here we use some ideas from [15]. It follows from (i) that

H^{m−k}(M) = Im L + P_{m−k},

where P_{m−k} = {a ∈ H^{m−k}(M) | L^{k+1}a = 0}. Indeed, if a ∈ H^{m−k}(M) then there exists b ∈ H^{m−k−2}(M) such that L^{k+1}a = L^{k+2}b. Therefore, a − b ∧ [ω] ∈ P_{m−k}, and

a = b ∧ [ω] + (a − b ∧ [ω]) ∈ Im L + P_{m−k}.

Because of (ii), every class in Im L is symplectically harmonic. So, it suffices to prove that any cohomology class a ∈ P_{m−k} is symplectically harmonic.

Let a = [α] ∈ P_{m−k} with α ∈ Ω^{m−k}(M) closed. Since L^{k+1}a = 0 ∈ H^{m+k+2}(M), there exists β ∈ Ω^{m+k+1}(M) such that α ∧ ω^{k+1} = dβ. It is known [8] that L^{k+1} : Ω^{m−k−1}(M) ⟶ Ω^{m+k+1}(M) is surjective. So, there exists γ ∈ Ω^{m−k−1}(M) with β = γ ∧ ω^{k+1}, and hence (α − dγ) ∧ ω^{k+1} = 0. So, if we take \overline{α} = α − dγ, then [\overline{α}] = a and L^{k+1}\overline{α} = 0. But the equality L^{k+1}\overline{α} = 0 implies that L^{∗}\overline{α} = 0 in view of (2.2). Thus, \overline{α} is

symplectically harmonic by (2.1).  QED

Corollary 3.2 ([15]) Let (M, ω) be an arbitrary symplectic manifold. Then every cohomology class in H^{k}(M), k = 0, 1, 2, is symplectically harmonic. □

Recall that a symplectic manifold (M^{2m}, ω) is called a manifold of Lefschetz type, if the map

L : H^{1}(M) ⟶ H^{2m−1}(M)

is surjective. Notice that, similarly to 2.15, b_{1}(M) is even for every closed manifold M of Lefschetz type.

Corollary 3.3 Let (M^{2m}, ω) be a manifold of Lefschetz type. Then every cohomology class in H^{3}(M) is symplectically harmonic. □

Corollary 3.4 For every symplectic manifold (M^{2m}, ω) and k = 0, 1, 2,

H_{hr}^{2m−k}(M) = Im{L^{m−k} : H^{k}(M) → H^{2m−k}(M)} ⊂ H^{2m−k}(M).

Furthermore, if M is a Lefschetz type manifold then

H_{hr}^{2m−3}(M) = Im{L^{m−3} : H^{3}(M) → H^{2m−3}(M)} ⊂ H^{2m−3}(M).
Proof : This follows from 2.7 because of 3.2 and 3.3.  QED

Corollary 3.5 For every closed symplectic manifold (M^{2m}, ω) the number h_{2m−1}(M^{2m}, ω) is even. Furthermore, if b_{2} = 1 then h_{2m−1}(M^{2m}, ω) does not depend on ω.

Proof : It follows from 2.14 and 2.17 since, by 3.4, ρ_{1} = h_{2m−1}.  QED

Corollary 3.6 If (M^{2m}, ω) is a symplectic manifold that is not a manifold of Lefschetz type, then h_{2m−1} ≤ b_{1} − 1. In particular, if b_{1} = 2, then h_{2m−1} = 0. □

This is the case for compact non-toral nilmanifolds (see [14] and the table of the classification of 6-dimensional compact nilmanifolds in Section 5).

4 Yan’s result on flexibility in dimension 4

According to Yan [15], 4-dimensional compact nilmanifolds are not flexible. Indeed, by 3.2, only h_{3} may vary, but it turns out that h_{3} is constant. Namely, based on certain results from [4], Yan [15] noticed the following relation for closed 4-dimensional nilmanifolds:

(i) if b_{1}(M) = 2, then h_{3} = 0,

(ii) if b_{1}(M) = 3 (therefore, b_{2}(M) = 4), then h_{3} = 2,

(iii) if b_{1}(M) = 4 (i.e. M = 𝕋^{4}), then h_{3} = 4.

On the other hand, Yan [15] has found closed 4-dimensional flexible manifolds, although his arguments need a certain correction (see below). Namely, he formulated without proof the following proposition, where M is assumed to be a closed 4-dimensional manifold.

Proposition 4.1 ([15, Prop. 4.1]) The following assertions are equivalent:

(i) There exists a family ω_{t} of symplectic forms such that h_{3} varies.

(ii) There exist two symplectic forms ω_{1} and ω_{2} such that Im L_{[ω_{1}]} ≠ Im L_{[ω_{2}]}, where L_{[ω_{i}]} is the Lefschetz map respect to [ω_{i}], (i = 1, 2).

(iii) There exists a symplectic form ω on M and a class a ∈ H^{2}(M) such that Im L_{a} ⊄ Im L_{[ω]}.

(iv) There exists a symplectic form ω on M such that Im L_{[ω]} is not equal to the image of the cup product pairing H^{1}(M) ⊗ H^{2}(M) ⟶ H^{3}(M). □

Concerning to this proposition, it is true that (i)⇒ (ii) ⇒ (iii) ⇒ (iv), but the Kodaira– Thurston manifold satisfies (iv) and does not satisfy (i). The Kodaira–Thurston manifold is obtained by taking the product of the Heisenberg manifold and the circle (this manifold is a compact nilmanifold). Its Sullivan minimal model has the form

(Λ(x_{1}, x_{2}, x_{3}, x_{4}), d) with dx_{1} = dx_{2} = dx_{4} = 0, dx_{3} = x_{1}x_{2},

where deg x_{i} = 1 for i = 1, 2, 3, 4 and the generators x_{1}, x_{2}, x_{3} come from the Heisenberg manifold. The cohomology class of the symplectic form is then given by the element ω = x_{1}x_{4} + x_{2}x_{3}. Now, from a direct computation we obtain that Im L_{[ω]} is generated by the non-zero classes of x_{1}x_{2}x_{3} and x_{2}x_{3}x_{4} and the image of the cup product H^{1}(M) ⊗ H^{2}(M) ⟶ H^{3}(M) is generated by x_{1}x_{2}x_{3}, x_{1}x_{3}x_{4} and x_{2}x_{3}x_{4}, so condition (iv) is satisfied. Moreover, it is easy to see that the Kodaira–Thurston manifold satisfies the condition (ii).

But, because of what we have said in the beginning of the section, any 4-dimensional nilmanifold (and hence the Kodaira-Thurston manifold) does not satisfy condition (i).

Yan’s construction of flexible closed 4-dimensional manifolds is based on the following proposition.

Proposition 4.2 [15, Cor. 4.2] Let (M^{4}, ω) be a closed symplectic manifold which satisfies the following conditions:

(i) the homomorphism L_{[ω]} : H^{1}(M) ⟶ H^{3}(M) is trivial;

(ii) the cup product H^{1}(M) ⊗ H^{2}(M) ⟶ H^{3}(M) is non-trivial.

Then M is flexible. □

Yan regards this proposition as a corollary of his Proposition 4.1. As we have seen, the last one is wrong. However, Proposition 4.2 is correct because it is a special case of our Corollary 2.12.

Finally, Gompf [7, Observation 7] proved the existence of 4-manifolds as in 4.2.

5 Symplectically harmonic forms in homogeneous spaces

From the previous section we know that none 4-dimensional nilmanifold is flexible. The goal of this section is to demonstrate the existence of 6-dimensional flexible nilmanifolds.

A compact nilmanifold is a homogeneous space of the form G/Γ, where G is a simply connected nilpotent Lie group and Γ is a discrete co-compact subgroup of G, i.e. a lattice (co-compact means that G/Γ is compact). Recall that Γ is determined by G uniquely up to an isomorphism. In greater detail, if Γ and Γ′ are two lattices in G then there exists an automorphism φ : G → G with φ(Γ) = Γ′, see [11]. Moreover, Γ determines G uniquely up to an isomorphism. In particular, the compact nilmanifold G/Γ determines and is completely determined by G.

Three important facts in the study of compact nilmanifolds are (see [14]):

(i) Let 𝔤 be a nilpotent Lie algebra with structural constants c_{k}^{ij} with respect to some basis, and let {α_{1}, …, α_{n}} be the dual basis of 𝔤^{∗}. Then in the Chevalley–Eilenberg complex (Λ^{∗}𝔤^{∗}, d) we have

dα_{k} = ∑_{1≤i<j<k} c_{k}^{ij}α_{i} ∧ α_{j}.  (5.1)

(ii) Let 𝔤 be the Lie algebra of a simply connected nilpotent Lie group G. Then, by Malcev’s theorem, G admits a lattice if and only if 𝔤 admits a basis such that all the structural constants are rational.

(iii) By Nomizu’s theorem, the Chevalley–Eilenberg complex (Λ^{∗}𝔤^{∗}, d) of 𝔤 is quasiisomorphic to the de Rham complex of G/Γ. In particular,

H^{∗}(G/Γ) ≅ H^{∗}(Λ^{∗}𝔤^{∗}, d)  (5.2)

and any cohomology class [a] ∈ H^{k}(G/Γ) contains a homogeneous representative α. Here we call the form α homogeneous if the pullback of α to G is left invariant.

Theorem 5.1 There exist at least five 6-dimensional flexible nilmanifolds.

To prove this theorem we run our fingers over all 34 6-dimensional compact nilmanifolds. The results are contained in the table below. The proofs take all the remained part of the section.

It follows from Corollary 3.2 that dim H_{hr}^{k}(M) = dim H^{k}(M), for k ≠ 3, 4, 5. Therefore, we should study the behaviour of h_{k} = dim H_{hr}^{k}(M) for degrees k = 3, 4, 5. We were not able to compute h_{3}, but we have found 5 manifolds with h_{4} and/or h_{5} varying. So, by 2.11, there are at least five 6-dimensional flexible nilmanifolds. (There are some reasons to conjecture that h_{3} = b_{3} for all closed 6-dimensional manifolds. So, if it is true then we have exactly five 6-dimensional flexible nilmanifolds.)

It turns out that every 6-dimensional real nilpotent Lie algebra admits a basis with rational structural constants. So, by what we said above, the 6-dimensional compact nilmanifolds are in a bijective correspondence with the 6-dimensional simply connected nilpotent Lie groups, and hence with the 6-dimensional nilpotent Lie algebras.

We use the classification of nilpotent Lie algebras given by Salamon [12]. It is based on the Morozov classification of 6-dimensional nilpotent Lie algebras [10]. We have added to Salamon’s classification the symplectically harmonic Betti numbers h_{k}(M) for k = 4, 5.

In the table Lie algebras appear lexicographically with respect to (b_{1}, b_{2}, 6 − s) where s is the step length. The first two columns contain the Betti numbers b_{1} and b_{2} (notice that b_{3} = 2(b_{2} − b_{1} + 1) because of the vanishing of the Euler characteristic). The next column contains 6 − s, where s is the step length.

The fourth column contains the description of the structure of the Lie algebra by means of the expressions of the form (5.1) in the Chevalley-Eilenberg complex. In view of 5.2, it means that, say, for the compact nilmanifold M from the second row, there exists a basis {α_{i}}_{i=1}^{6} of homogeneous 1-forms on M such that

dα_{1} = 0 = dα_{2}, dα_{3} = α_{1} ∧ α_{2}, dα_{4} = α_{1} ∧ α_{3}, dα_{5} = α_{1} ∧ α_{4}, dα_{6} = α_{3} ∧ α_{4} + α_{5} ∧ α_{2}.

The column headed ⊕ indicates the dimensions of the irreducible subalgebras in case 𝔤 is not itself irreducible.

The next columns show the dimensions h_{k} for k = 4, 5. So, the column, say, h_{4} contains all possible values of h_{4}(M, ω) which appear when ω runs over all symplectic forms on M. The sign “–” at a certain row means that the corresponding Lie algebra (as well as the compact nilmanifold) does not admit a symplectic structure.

For completeness, in the last columns we list the real dimension dimℝS(𝔤) of the moduli space of symplectic structures.

Convention 5.2 (i) From now on we write α_{ij··· k} instead of α_{i} ∧ α_{j} ∧ ⋯ ∧ α_{k}.

(ii) In future we say that a compact nilmanifold G/Γ has type, say (0,0,12,13,14,15) if the corresponding Lie algebra has the structure (0,0,12,13,14,15) (i.e. in our case, sits in the third row).

Six-dimensional real nilpotent Lie algebras

b_{1} b_{2} 6−s  Structure  ⊕  h_{4} h_{5} dimℝS(𝔤)
2 2 1 (0, 0, 12, 13, 14 + 23, 34 + 52)  –  –  –
2 2 1 (0, 0, 12, 13, 14, 34 + 52)  –  –  –
2 3 1 (0, 0, 12, 13, 14, 15)  3  0  7
2 3 1 (0, 0, 12, 13, 14 + 23, 24 + 15)  2  0  7
2 3 1 (0, 0, 12, 13, 14, 23 + 15)  2  0  7
2 4 2 (0, 0, 12, 13, 23, 14)  4  0  8
2 4 2 (0, 0, 12, 13, 23, 14 − 25)  2, 3, 4 0  8
2 4 2 (0, 0, 12, 13, 23, 14 + 25)  4  0  8
3 4 2 (0, 0, 0, 12, 14 − 23, 15 + 34)  2  0  7
3 5 2 (0, 0, 0, 12, 14, 15 + 23)  4  2  8
3 5 2 (0, 0, 0, 12, 14, 15 + 23 + 24)  3, 4 0, 2  8
3 5 2 (0, 0, 0, 12, 14, 15 + 24)  1 + 5  4  2  8
3 5 2 (0, 0, 0, 12, 14, 15)  1 + 5  4  2  8
3 5 3 (0, 0, 0, 12, 13, 14 + 35)  –  –  –
3 5 3 (0, 0, 0, 12, 23, 14 + 35)  –  –  –
3 5 3 (0, 0, 0, 12, 23, 14 − 35)  –  –  –
3 5 3 (0, 0, 0, 12, 14, 24)  1 + 5  –  –  –
3 5 3 (0, 0, 0, 12, 13 + 42, 14 + 23)  3  0  8
3 5 3 (0, 0, 0, 12, 14, 13 + 42)  3  0  8
3 5 3 (0, 0, 0, 12, 13 + 14, 24)  2, 3 0  8
3 6 3 (0, 0, 0, 12, 13, 14 + 23)  3  0  9
3 6 3 (0, 0, 0, 12, 13, 24)  5  0  9
3 6 3 (0, 0, 0, 12, 13, 14)  4  0  9
3 8 4 (0, 0, 0, 12, 13, 23)  7, 8 0  9
4 6 3 (0, 0, 0, 0, 12, 15 + 34)  –  –  –
4 7 3 (0, 0, 0, 0, 12, 15)  1+1+4  3  2  9
4 7 3 (0, 0, 0, 0, 12, 14 + 25)  1 + 5  3  2  9
4 8 4 (0, 0, 0, 0, 13 + 42, 14 + 23)  7  2  10
4 8 4 (0, 0, 0, 0, 12, 14 + 23)  6  2  10
4 8 4 (0, 0, 0, 0, 12, 34)  3 + 3  7  2  10
4 9 4 (0, 0, 0, 0, 12, 13)  1 + 5  7, 8 2  11
5 9 4 (0, 0, 0, 0, 0, 12 + 34)  1 + 5  –  –  –
5 11 4 (0, 0, 0, 0, 0, 12)  1+1+1+3 9  4  12
6 15 5 (0, 0, 0, 0, 0, 0)  1 + ⋯ + 1 15 6  15

Proof of Theorem 5.1

We prove the theorem via considering case by case. Namely, we study in more detail the cases which are proclaimed to be flexible. In view of 2.11, they are precisely the cases of compact nilmanifolds with varying symplectically harmonic Betti numbers h_{k}. Here the main tool for computing h_{k} is Corollary 3.4. In order to find symplectic structures on M, we use the following proposition.

Proposition 5.3 Let M^{2n} be a compact manifold of the form G/Γ where Γ is a discrete subgroup of a Lie group G, and let ω ∈ Ω^{2}(M) be a closed homogeneous 2-form such that [ω]^{n} ≠ 0. Then ω is a symplectic form on M.

Proof : Since [ω]^{n} ≠ 0, we conclude that the linear form ω|T_{x}M is non-degenerate for some point x ∈ M. So, ω^{n} is non-degenerate since it is homogeneous. Thus, ω is nondegenerate.

QED

Proposition 5.4 The compact nilmanifold M of the type (0, 0, 12, 13, 23, 14 − 25) is flexible.

Proof : According to our assumption about the type of M, there exists a basis {α_{i}}_{i=1}^{6} of homogeneous 1-forms on M such that

dα_{1} = dα_{2} = 0, dα_{3} = α_{12}, dα_{4} = α_{13}, dα_{5} = α_{23}, dα_{6} = α_{14} − α_{25}.

Since the de Rham cohomology of the nilmanifold is isomorphic to the Chevalley-Eilenberg cohomology of the Lie algebra, we conclude that

H^{1}(M) = {[α_{1}], [α_{2}]},
H^{2}(M) = {[α_{14}], [α_{15} + α_{24}], [α_{26} − α_{34}], [α_{16} − α_{35}]}.

In particular, by 5.3, a 2-form ω on M is symplectic if and only if

[ω] = A[α_{14}] + B[α_{15} + α_{24}] + C[α_{26} − α_{34}] + D[α_{16} − α_{35}],  (5.3)

where ACD − B(C^{2} + D^{2}) ≠ 0, A, B, C, D ∈ ℝ.

Claim 5.5 Let ω be a symplectic form on M. Then the following holds:

(i) if C^{2} ≠ D^{2}, then h_{4} = 4;

(ii) if C^{2} = D^{2} and A^{2} ≠ 4B^{2}, then h_{4} = 3;

(iii) if C^{2} = D^{2} and A^{2} = 4B^{2}, then h_{4} = 2.

Furthermore, h_{5} = 0 for every symplectic form ω on M.

Proof : It follows from the Poincaré duality that H^{4}(M) = ℝ^{4}. Hence, by (5.2)

H^{4}(M) = {[α_{1246}], [α_{1256}], [α_{1356}], [α_{1346} + α_{2356}]}

since the four cohomology classes from above are linearly independent. Furthermore, for every ω the image of the mapping L : H^{2}(M) ⟶ H^{4}(M) is

Im L = {−C[α_{1246}] + D[α_{1256}], 2D[α_{1246}] − 2C[α_{1256}],
−A[α_{1246}] − 2B[α_{1256}] − 2C[α_{1356}] − D[α_{1346} + α_{2356}],
2B[α_{1246}] + A[α_{1256}] − 2D[α_{1356}] − C[α_{1346} + α_{2356}]},

which has dimension 4 for C^{2} ≠ D^{2}, dimension 3 for C^{2} = D^{2} and A^{2} ≠ 4B^{2}, and

dimension 2 for C^{2} = D^{2} and A^{2} = 4B^{2}. The result follows from Corollary 3.4.  QED
Now, the proof of the proposition follows from 2.11.  QED

Proposition 5.6 The compact nilmanifold of the type (0, 0, 0, 12, 14, 15 + 23 + 24) is flexible.

Proof : According to our assumption about the type of M, there exists a basis {α_{i}}_{i=1}^{6} of homogeneous 1-forms on M such that

dα_{1} = dα_{2} = dα_{3} = 0, dα_{4} = α_{12}, dα_{5} = α_{14}, dα_{6} = α_{15} + α_{23} + α_{24}.

The cohomology groups of degrees 1, 2 are:

H^{1}(M) = {[α_{1}], [α_{2}], [α_{3}]},
H^{2}(M) = {[α_{13}], [α_{15}], [α_{23}], [α_{16} + α_{25} − α_{34}], [α_{26} − α_{45}]}.

In particular, if ω is a symplectic form on M then

[ω] = A[α_{13}] + B[α_{15}] + C[α_{23}] + D[α_{16} + α_{25} − α_{34}] + E[α_{26} − α_{45}],  (5.4)

where AE^{2} + BDE − CDE − D^{3} ≠ 0.

The following claim can be proved similarly to 5.5.

Claim 5.7 Let ω be a symplectic form on M. Then the following holds:

(i) if E ≠ 0, then h_{4} = 4, h_{5} = 2;

(ii) if E = 0, then h_{4} = 3, h_{5} = 0. □

Now, by 2.11, M is flexible.  QED

We hope that now it is clear how to run over all the three remaining cases. So, below we omit the details while indicate the main steps of the corresponding calculations.

Proposition 5.8 The compact nilmanifolds of types (0, 0, 0, 12, 13 + 14, 24), (0, 0, 0, 12, 13, 23) and (0, 0, 0, 0, 12, 13) are flexible.

Proof : \underline{Case (0, 0, 0, 12, 13 + 14, 24):}

H^{2}(M) = {[α_{13}], [α_{15}], [α_{23}], [α_{16} + α_{25} + α_{34}], [α_{26}]},  (5.5)

and the 2-form ω on M is symplectic if and only if

[ω] = A[α_{13}] + B[α_{15}] + C[α_{23}] + D[α_{16} + α_{25} + α_{34}] + E[α_{26}],

where D(BE − D^{2}) ≠ 0. Furthermore, if EB + 3D^{2} ≠ 0 then h_{4} = 3; otherwise, h_{4} = 2. Finally, h_{5} = 0 for every ω. \underline{Case (0, 0, 0, 12, 13, 23):}

H^{2}(M) = {[α_{14}], [α_{15}], [α_{16} + α_{25}], [α_{16} − α_{34}], [α_{24}], [α_{26}], [α_{35}], [α_{36}]}, and the 2-form ω on M is symplectic if and only if [ω] = A[α_{14}] + B[α_{15}] + C[α_{16} + α_{25}] + D[α_{16} − α_{34}] + E[α_{24}] + F [α_{26}] + G[α_{35}] + H[α_{36}], where

ACH − AF G − BDF − BEH + DC^{2} + CEG + CD^{2} + DEG ≠ 0.

Furthermore, if C^{2} + CD + D^{2} − BF − EG + AH ≠ 0, then h_{4} = 8; otherwise, h_{4} = 7. Finally, h_{5} = 0 for every ω. \underline{Case (0, 0, 0, 0, 12, 13):} The second de Rham cohomology group is given by

H^{2}(M) = {[α_{14}], [α_{15}], [α_{16}], [α_{23}], [α_{24}], [α_{25}], [α_{34}], [α_{26} + α_{35}], [α_{36}]}. The 2-form ω on M is symplectic if and only if

[ω] = A[α_{14}] + B[α_{15}] + C[α_{16}] + D[α_{23}] + E[α_{24}] + F [α_{25}]
+ G[α_{34}] + H[α_{26} + α_{35}] + I[α_{36}],

where −AF I + H^{2}A + BEI − BGH − CEH + CF G ≠ 0. Furthermore, if H^{2} − F I ≠ 0, then h_{4} = 8; otherwise, h_{4} = 7. Finally, h_{5} = 2 for every ω. Notice that in this last case the nilpotent Lie algebra (so the compact nilmanifold) is

reducible of type (1 + 5).  QED

Comment 5.9 Here we want to say more about flexibility of manifolds appeared in 5.4 and 5.6.

(a) Consider the manifold from 5.4 and the family

ω_{t} = −\frac{t}{2}(t − 3)α_{14} + \frac{1}{4}(t^{2} − 5t + 4)(α_{15} + α_{24}) − \frac{t}{2}(t − 3)(α_{26} − α_{34}) + α_{16} − α_{35}

of closed 2-forms on M. The form ω_{t} is symplectic if

t^{6} − 11t^{5} + 39t^{4} − 45t^{3} + 4t^{2} − 20t + 16 ≠ 0.

This polynomial has two real roots, and both of them lie out of the interval (1 − ε, 4). So, because of 5.5

h_{4}(ω_{2}) = 2, h_{4}(ω_{1}) = h_{4}(ω_{\frac{3+\sqrt{17}}{2}} ) = 3,

and h_{4}(ω_{t}) = 4 for all other t ∈ (1 − ε, 4).

(b) Consider the manifold from 5.6 and the family

ω_{t} = (1 − t)α_{13} − t(α_{16} + α_{25} − α_{34}) + (1 − t)(α_{26} − α_{45})

of closed 2-forms. Since the polynomial AE^{2} + BDE − CDE − D^{3} = 3t^{2} − 3t + 1 has no real roots, we conclude that ω_{t} is a family of symplectic structures and

(i) h_{4}(ω_{1}) = 3, h_{5}(ω_{1}) = 0;

(ii) h_{4}(ω_{t}) = 4, h_{5}(ω_{t}) = 2 for t ≠ 1.

Remark 5.10 Minding Theorem 5.1, it is natural to ask whether there exist flexible nilmanifolds of dimension greater than 6. Taking into account the results of the next section, we see that the answer is affirmative. However, the question remains open for irreducible compact nilmanifolds of dimension greater than 6.

Remark 5.11 We have also considered the 6-dimensional compact completely solvable manifold M constructed by Fernández–de León–Saralegi [6]. This compact manifold does not satisfy the Hard Lefschetz condition (although it is of Lefschetz type). We have

b_{1}(M) = b_{5}(M) = 2, b_{2}(M) = b_{4}(M) = 3, b_{3}(M) = 4.

Furthermore, h_{3}(M, ω) = 4 and h_{4}(M, ω) = h_{5}(M, ω) = 2 for every symplectic form ω on M. We do not explain the details because M is not flexible.

The following question also seems to be interesting.

Question: Is there a Nomizu’s type result for compact nilmanifolds (more generally, homogeneous spaces) and the symplectically harmonic cohomology H_{hr}^{∗}(M). In another words, does a symplectically harmonic de Rham cohomology class contain a homogeneous symplectically harmonic representative (if we are considering homogeneous symplectic structures)?

The answer is affirmative for degrees k ≤ 2 and k ≥ 2m − 2. Indeed, let G/Γ be a 2mdimensional compact nilmanifold with a homogeneous symplectic form ω, and let 𝔤 be the Lie algebra of G. Since the image of a homogeneous form under each of the operators ∗, d and L is homogeneous, the (finite dimensional) subspaces Ω_{hr}^{∗}(𝔤^{∗}) and Λ^{∗}(𝔤^{∗}) are 𝔰𝔩(2)-submodules of Ω^{∗}(G/Γ). Therefore, obvious analogs of Proposition 2.5, Corollaries 3.2 and 3.4 hold for 𝔤^{∗} and the result follows from the Nomizu’s theorem.

6 Product formula for symplectically harmonic cohomology

Let (M^{2m}, ω_{1}) and (N^{2n}, ω_{2}) be two symplectic manifolds. Consider the symplectic product manifold (M × N, ω) where ω = p_{1}^{#}ω_{1} + p_{2}^{#}ω_{2} and

p_{1} : M × N → M, p_{2} : M × N → N

are the projections. Given two forms α ∈ Ω^{p}(M) and β ∈ Ω^{q}(N), consider the form

α ⊠ β := (p_{1}^{#}α) ∧ (p_{2}^{#}β) ∈ Ω^{p+q}(M × N).

Proposition 6.1 ([2])

∗(α ⊠ β) = (−1)^{pq}(∗_{1}α) ⊠ (∗_{2}β),
□

Corollary 6.2 (i) i(Π)(α ⊠ β) = (i(Π_{1})α) ⊠ β + α ⊠ (i(Π_{2})β);

(ii) δ(α ⊠ β) = (δ_{1}α) ⊠ β + (−1)^{p}α ⊠ (δ_{2}β);

(iii) Ω_{hr}^{p}(M) ⊠ Ω_{hr}^{q}(N) ⊂ Ω_{hr}^{p+q}(M × N);

(iv) for all k we have

∑_{p+q=k} h_{p}(M)h_{q}(N) ≤ h_{k}(M × N).

□

Question: When the inequality in 6.2(iv) turns out to be the equality?

Now we consider the Lefschetz map L_{[ω]}^{m+n−k} : H^{k}(M × N) ⟶ H^{2m+2n−k}(M × N).

Proposition 6.3

h_{2(m+n)−1}(M × N) = h_{2m−1}(M) + h_{2n−1}(N),
h_{2(m+n)−2}(M × N) = h_{2m−2}(M) + h_{2m−1}(M)h_{2n−1}(N) + h_{2n−2}(N).

Proof : Because of the Künneth isomorphism H^{∗}(M × N) ≅ H^{∗}(M) ⊗ H^{∗}(N), we conclude that

Im (L_{[ω]}^{m+n−1}) = [ω_{1}]^{m} ⊗ (Im L_{[ω_{2}]}^{n−1}) ⊕ (Im L_{[ω_{1}]}^{m−1}) ⊗ [ω_{2}]^{n},

and the first equality follows from 3.4. Similarly,

Im (L_{[ω]}^{m+n−2}) = {([ω_{1}]^{m−1}u) ⊗ [ω_{2}]^{n−1} + ([ω_{1}]^{m−2}u) ⊗ [ω_{2}]^{n} | u ∈ H^{2}(M)}
⊕ {[ω_{1}]^{m} ⊗ ([ω_{2}]^{n−2}v) + [ω_{1}]^{m−1} ⊗ ([ω_{2}]^{n−1}v) | v ∈ H^{2}(N)}
⊕ {([ω_{1}]^{m−1}w_{1}) ⊗ ([ω_{2}]^{n−1}w_{2}) | w_{1} ∈ H^{2}(M), w_{2} ∈ H^{2}(N)}.
Now, in view of 3.4, the computation of dimensions completes the proof.  QED

Corollary 6.4 Let M^{2m} be a manifold which admits a family of symplectic forms such that the symplectically harmonic Betti number h_{2m−k} varies for k = 1 or k = 2. Then M × N is a flexible manifold whenever a manifold N admits a symplectic structure.

Proof : If h_{2m−1}(M) varies then the result follows from the first equality of 6.3. If h_{2m−2}(M) varies but h_{2m−1}(M) does not vary then the result follows from the second

equality of 6.3.  QED

7 Duality

Consider a symplectic manifold (M^{2m}, ω) and the chain complex

⋯ ⟶ Ω^{k+1}(M) ⟶^{δ} Ω^{k}(M) ⟶^{δ} Ω^{k−1}(M) ⟶ ⋯

with δ as in (2.1). The following proposition follows directly from the definition of δ.

Proposition 7.1 (i) δα = 0 if and only if d ∗ α = 0;

(ii) α ∈ Im δ if and only if ∗α ∈ Im d. □

We define

H_{δ}^{k}(M) = H_{δ}^{k}(M, ω) = Ker δ^{k}/Im δ^{k+1} where δ^{i} = δ : Ω^{i} → Ω^{i−1}.

Corollary 7.2 The operator ∗ : Ω^{k} → Ω^{2m−k} induces an isomorphism

∗ : H^{k}(M) → H_{δ}^{2m−k}(M).

In particular, H_{δ}^{k}(M) = H^{k}(M) for M closed. □

We dualize the definition of symplectically harmonic Betti numbers h_{k} by setting

h_{k}^{∗}(M) = h_{k}^{∗}(M, ω) := dim (Ω_{hr}^{k}/Im δ ∩ Ω_{hr}^{k})

Corollary 7.3 h_{m−k}^{∗}(M) = h_{m+k}(M). □

In particular, in view of 3.2, if M is closed then h_{2m−k}^{∗}(M) = b_{2m−k}(M) for k = 0, 1, 2.

It is clear that many other results of Sections 2 and 3 can be dualized in a similar way. We leave it to the reader.

Acknowledgments

This work was done during the stay of three of the authors in the Mathematisches Forschungsinstitut in Oberwolfach under the “Research in Pairs” program financed by the “Volkswagen-Stiftung”. We express our sincere thanks to the Institute and to the Foundation for the hospitality and wonderful atmosphere. The third author was partially financed by the Polish Research Commitee (KBN). The first and fourth authors were partially supported by the project UPV 127.310-EA147/98 and DGICYT PB97-0504-C02-02.

We thank Robert Gompf, Boris Khesin and Dusa McDuff for helpful discussions.

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