Transversely Kähler foliations

∆∂¯:= ¯∂ ¯∂^∗+ ¯∂^∗∂.¯

With the help of this operators one can state and prove the Hodge decomposition for basic and basic Dolbeault cohomology. Furthermore, in the transversely Kähler case the kernels of these operators are equal (cf. [16]). Since α is d-exact, it is orthogonal to the space of ∆-harmonic forms. Hence it is also orthogonal to the space of ∆∂-harmonic forms. Since α is also ∂-closed, it has to be ∂-exact as well (by the Hodge decomposition for basic Dolbeault cohomology). Let β be a basic form such that α = ∂β. By applying the Hodge decomposition again we get β = h + ¯∂η + ¯∂^∗ξ, where h is

∆∂¯-harmonic (h is also ∆∂-harmonic). It suffices to prove that ¯∂^∗ξ is ∆∂-harmonic. Since α is

∂-closed, we get ¯¯ ∂ ¯∂^∗∂ξ = 0 (we use here the identity ∂ ¯∂^∗+ ¯∂^∗∂ = 0 proven for Kähler foliations in [16]). Using the scalar product associated to the transverse Riemannian metric we get:

|| ¯∂^∗∂ξ||²=< ∂ξ, ¯∂ ¯∂^∗∂ξ >= 0.

This means that α = ∂ ¯∂η.

4.4. TRANSVERSELY KÄHLER FOLIATIONS 41 Let us now prove the dd^Λ-lemma for transversely Kähler homologically orientable foliations:

Theorem 4.21. Let F be a transversely Kähler homologically orientable foliation on a compact manifold M. Then the following statements are true:

1. F satisfies the basic dd^Λ-lemma.

2. i : (Ker(d^Λ), d) → (Ω^•(M/F ), d) induces an isomorphism in cohomology.

3. The subbundles U^k induce a decomposition in cohomology.

4. p : (Ω^•(M/F , C), d^Λ) → (Ω^•(M/F , C)/Im(d), d^Λ) induces an isomorphism in cohomology.

5. The Lefschetz map L^k : H^q−k(M/F ) → H^q+k(M/F ) is surjective (or equivalently bijective by the foliated version of Poincaré Duality).

6. Every d-closed form has a d^Λ-closed representative.

Proof. The fifth condition is satisfied under our assumptions (cf. [16]). The rest follows from Corollary 4.19.

Chapter 5

Transverse Frölicher type inequalities

The purpose of this section is to generalize the Frölicher type inequalities (proved in [4, 5]) to basic cohomology (as was done in [32]). These inequalities provide a computable way of verifying whether a given foliation satisfies the basic ∂ ¯∂-lemma or the dd^Λ-lemma by comparing the dimensions of certain special cohomology theoreis with the dimension of the basic cohomology. We also provide results from [14] which in a sense extend the relations given in the formerly mentioned inequalities to the foliations with infinitely dimensional basic cohomology.

5.1 General Frölicher type inequalities

Let (K^•,•, d⁰, d⁰⁰) be a double cochain complex of modules with d⁰ : K^•,•→ K^•+1,•and d⁰⁰: K^•,•→ K^•,•+1, and let D := d⁰+ d⁰⁰ be its total coboundary operator. We say that K^•,• satisfies the d⁰d⁰⁰-lemma iff:

Ker(d⁰) ∩ Ker(d⁰⁰) ∩ Im(D) = Im(d⁰d⁰⁰).

The first theorem stated here will give us a couple of conditions equivalent to satysfying the d⁰d⁰⁰ -lemma.

Theorem 5.1. ([15], Lemma 5.15) Let K^•,• be a bounded double complex (i.e. such that for each n ∈ N almost all K^i,j with i+j = n are zero) with notation as above. Then the following conditions are equivalent:

• K^•,• satisfies the d⁰d⁰⁰-lemma.

• Ker(d⁰⁰) ∩ Im(d⁰) = Im(d⁰d⁰⁰) and Ker(d⁰) ∩ Im(d⁰⁰) = Im(d⁰d⁰⁰).

• Ker(d⁰) ∩ Ker(d⁰⁰) ∩ (Im(d⁰) + Im(d⁰⁰)) = Im(d⁰d⁰⁰).

• Im(d⁰) + Im(d⁰⁰) + Ker(D) = Ker(d⁰d⁰⁰).

• Im(d⁰⁰) + Ker(d⁰) = Ker(d⁰d⁰⁰) and Im(d⁰) + Ker(d⁰⁰) = Ker(d⁰d⁰⁰).

• Im(d⁰) + Im(d⁰⁰) + (Ker(d⁰) ∩ Ker(d⁰⁰)) = Ker(d⁰d⁰⁰).

43

For any double chain complex there exist two filtrations of the associated total chain complex

This filtrations induce two spectral sequences ⁰E_r^p,q and⁰⁰E_r^p,q in the standard way (cf. [7]). This spectral sequences satisfy:

0E₀^p,q= K^{p,q 0}E₁^p,q = H_d^q00(K^p,•) ⁰E₂^p,q = H_d^p0(H_d^q00(K^•,•))

00E₀^p,q= K^{p,q 00}E₁^p,q= H_d^q0(K^•,p) ⁰⁰E₂^p,q= H_d^p00(H_d^q0(K^•,•)).

Furthermore, if K^•,• is a first (or third) quadrant double complex, then both associated spectral sequences converge to the total cohomology of the complex K^•,•.

Theorem 5.2. ([15], Proposition 5.17) Let K^•,•be a bounded double complex of vector space. The d⁰d⁰⁰-lemma holds for K^•,• if and only if the two spectral sequences associated to K^•,• degener-ate at the first page and are k-opposite for k ∈ N (i.e., ⁰F^p(T ot^k(K^•,•)) ⊕ ⁰⁰F^q(T ot^k(K^•,•)) ∼= H_D^k(T ot(K^•,•)) for p + q − 1 = k).

We proceed to state two main theorems from the paper [4] concerning the Frölicher-type in-equality for graded and bigraded vector spaces. We will use these abstract theorems to establish the Frölicher type inequalities for transversely holomorphic and transversely symplectic foliations (the non-foliated case was treated in [3] and [4]). We will also present a proposition in similar spirit to [14] and some of its consequences. Let us start with a bigraded K-vector space V^•,• endowed with two endomorphisms d⁰ and d⁰⁰ with order (d⁰₁, d⁰₂) and (d⁰⁰₁, d⁰⁰₂) respectively (and let us denote as before D = d⁰+ d⁰⁰). Furthermore, let us assume that this endomorphisms satisfy the condition:

(d⁰)²= (d⁰⁰)²= d⁰d⁰⁰+ d⁰⁰d⁰= 0. (5.1) Let V^• = L

p+q=•

V^p,q denote the standard graded vector space asociated with V^•,•. We define the following cohomology of V^•,•:

H_D^•(V^•) := ^Ker(D)_Im(D) H_d^•,•0 (V^•,•) := ^Ker(d_Im(d0⁰)⁾

H_A^•,•(V^•,•) := _Im(d^Ker(d0)+Im(d^0d00)00) H_BC^•,•(V^•,•) := ^Ker(d_Im(d^0)∩Ker(d0d⁰⁰) ⁰⁰⁾.

We call them the total, d⁰, Aeppli and Bott-Chern cohomology, respectively. One can also define this cohomology groups in the graded case (i.e. for a graded vector space V^• endowed with two coboundary operators d⁰ and d⁰⁰which satisfy equation 5.1).

Theorem 5.3. Let V^•,• be as above. Suppose that:

dim_K(H_d^•00(V^•)) + dim_K(H_d^•0(V^•)) < ∞.

Then for every j ∈ Z the following inequality holds:

dim_K(H_BC^j (V^•)) + dim_K(H_A^j(V^•)) ≥ dim_K(H_d^j0(V^•)) + dim_K(H_d^j00(V^•)).

Furthermore, the equality holds iff V^•,• satisfies the d⁰d⁰⁰-lemma.

5.1. GENERAL FRÖLICHER TYPE INEQUALITIES 45 In order to state this theorem in the graded case we need one additional construction. Given a graded vector space V^•, endowed with two endomorphisms d⁰ and d⁰⁰ of order |d⁰| and |d⁰⁰|, respectively, satisfying (5.1) we define the associated bigraded vector space by:

Doub^p,q(V^•) := V^|d0|p+|d00|q.

We can also extend d⁰and d⁰⁰to this bigraded vector space in such a way that their orders are (1, 0) and (0, 1) respectively. With this we can now state the second main theorem of this subsection:

Theorem 5.4. Let (V^•, d⁰, d⁰⁰) be as above and let GCD(|d⁰|, |d⁰⁰|) = 1 and:

dim_K(H_d^•0(V^•)) + dim_K(H_d^•00(V^•)) < ∞.

Then for every j ∈ Z the following inequality holds:

dim_K(H_BC^j (V^•)) + dim_K(H_A^j(V^•)) ≥ dim_K(H_d^j0(V^•)) + dim_K(H_d^j00(V^•)). (5.2) Moreover, the following two conditions are equivalent:

1. The equality in (5.2) holds and the spectral sequences associated to the double complex Doub^•,•(V^•) degenerate at the first page.

2. V^• satisfies the d⁰d⁰⁰-lemma.

We finish this section by giving an interesting algebraic lemma from [14] which was omitted in [4], most likely due to limited use in the manifold case, however in the foliated case this result becomes far more potent as infinitely dimensional cohomology are encountered more often. Let I₁, I₂, I₁₂, K₁, K₂, K₁₂be vector spaces satisfying

Proof. Without loss of generality let us assume that K1

I₁ is infinite dimensional. There are two sequences

It is easy to see that these sequences are exact in the middle term (since the appropriate kernel and image are classes represented by elements of (K1∩ I2)). If both (K¹∩ K2) I12 and K¹².

(I1+ I2) have finite dimension, then so do (K1∩ I2) I₁₂ and K12

.

(I₁+ K₂) since they are smaller. But then the middle term has finite dimension by exactness, a contradiction.

This lemma in the context given above gives rise to the following theorems:

Theorem 5.6. If (V^•, d⁰, d⁰⁰) is as above and such that H_D^k(V^•) is infinite dimensional then H_BC^k (V^•) or H_A^k(V^•) is infinite dimensional as well.

Proof. In the lemma take:

K1= Ker(d⁰) K2= Ker(d⁰⁰) K12= Ker(d⁰d⁰⁰)

I1= Im(d⁰) I2= Im(d⁰⁰) I12= Im(d⁰d⁰⁰)

Theorem 5.7. If (V^•,•, d⁰, d⁰⁰) is as above and such that H_D^k(V^•) is infinite dimensional then H_BC^k (V^•) or H_A^k(V^•) is infinite dimensional as well.

Proof. In the lemma take:

K1= Ker(d⁰) K2= Ker(d⁰⁰) K12= Ker(d⁰d⁰⁰)

I1= Im(d⁰) I2= Im(d⁰⁰) I12= Im(d⁰d⁰⁰)