Cohomology of manifolds with structure group $U(n) \times O(s)$

Abstract

We introduce a new spectral sequence for the study of ${\mathcal {K}}$-manifolds which arises by restricting the spectral sequence of a Riemannian foliation to forms invariant under the flows of $\{\xi _1,\ldots ,\xi _s\}$. We use this sequence to generalize a number of theorems from K-contact geometry to ${\mathcal {K}}$-manifolds. Most importantly we compute the cohomology ring and harmonic forms of ${\mathcal {S}}$-manifolds in terms of primitive basic cohomology and primitive basic harmonic forms (respectively). As an immediate consequence of this we get that the basic cohomology of ${\mathcal {S}}$-manifolds are a topological invariant. We also show that the basic Hodge numbers of ${\mathcal {S}}$-manifolds are invariant under deformations. Finally, we provide similar results for ${\mathcal {C}}$-manifolds.

1 Introduction

In [19, 20] a study of manifolds with a tensor field f of type (1, 1) was initiated. The importance of the tensor field f stems from the fact that its existence is equivalent to the reduction of the structure group of the manifold to $U(n)\times O(s)$. As such this generalizes both the concept of almost complex and almost contact manifolds. The properties of the curvature of such manifolds were further studied in [5]. In particular, a special class of f-structures, called ${\mathcal {S}}$-structures, was introduced which generalizes Kähler and Sasakian manifolds. Since then these structures as well as their various generalizations (e.g. ${\mathcal {K}}$-structures, K-f-contact manifolds) were thoroughly studied.

The purpose of this article is to study the cohomological properties of such manifolds. In particular, we are interested in the relations between the cohomology of the manifold and the basic cohomology of the foliation defined by the ${\mathcal {S}}$-structure. This study is motivated by the important role played by basic cohomology in Sasakian Geometry (e.g. the Sasakian version of the Calabi-Yau Theorem) as well as new results from [7, 11]. We approach the problem by introducing and studying a new spectral sequence which is a variation of the spectral sequence of a Riemannian foliation (studied in e.g. [1, 2, 17]) and relates basic cohomology of a (almost) ${\mathcal {K}}$-manifold to its de Rham cohomology. Using this sequence we generalize a well known fact that for K-contact manifolds satisfying the hard Lefschetz property (equivalently the transverse hard Lefschetz property) the cohomology of the manifold in degree $r\le n$ is isomorphic to the primitive basic cohomology (see [14]). Due to Poincaré duality this allows us to recreate the cohomology of the manifold from basic primitive cohomology and viceversa, which implies that in such a case basic cohomology is a topological invariant. In fact, our results apply to a more general class of manifolds, namely almost ${\mathcal {S}}$-structures satisfying the basic hard Lefschetz property. Using similar methods we also prove an analogous result for ${\mathcal {C}}$-structures which constitutes another special class of manifolds distinguished in [5] and generalizing the notion of quasi-Sasakian manifolds with a closed 1-form $\eta$. An immediate corollary is that for manifolds with such structures the basic cohomologies are a topological invariant. In particular, this is true for any ${\mathcal {S}}$-manifolds and ${\mathcal {C}}$-manifolds.

We provide two additional applications of the above results. Firstly, we classify harmonic forms on ${\mathcal {S}}$-manifolds and ${\mathcal {C}}$-manifolds in terms of basic harmonic forms. This can be treated as a generalization of Proposition 7.4.13 from [6]. Secondly, we show that basic Hodge numbers of almost ${\mathcal {S}}$-manifolds and almost ${\mathcal {C}}$-manifolds which have the transverse hard Lefschetz property are invariant under deformations of such manifolds. This generalizes the main result from [16]. In particular, it is worth noting that this result applies to K-contact manifolds satisfying the hard Lefschetz property. Moreover, this partially answers Question 1.2 from [16], in that it gives a positiva answer to that question for a new class of transversely Kähler foliations.

We designate the subsequent section to preliminaries on basic cohomology and ${\mathcal {S}}$-structures. In Sect. 3 we describe the aforementioned spectral sequence which will be the key tool in this article. We apply it in Sects. 4 and 5 to prove the main results for almost ${\mathcal {S}}$-structures and almost ${\mathcal {C}}$-manifolds respectively. The final two sections contain the additional applications mentioned above.

2 Preliminaries

2.1 Foliations

We provide a quick review of transverse structures on foliations.

Definition 2.1

A codimension q foliation ${\mathcal {F}}$ on a smooth n-manifold M is given by the following data:

• An open cover ${\mathcal {U}}:=\{U_i\}_{i\in I}$ of M.

• A q-dimensional smooth manifold $T_0$.

• For each $U_i\in {\mathcal {U}}$ a submersion $f_i: U_i\rightarrow T_0$ with connected fibers (these fibers are called plaques).

• For all intersections $U_i\cap U_j\ne \emptyset$ a local diffeomorphism $\gamma _{ij}$ of $T_0$ such that $f_j=\gamma _{ij}\circ f_i$

The last condition ensures that plaques glue nicely to form a partition of M consisting of submanifolds of M of codimension q. This partition is called a foliation ${\mathcal {F}}$ of M and the elements of this partition are called leaves of ${\mathcal {F}}$.

We call $T=\coprod \limits _{U_i\in {\mathcal {U}}}f_i(U_i)$ the transverse manifold of ${\mathcal {F}}$. The local diffeomorphisms $\gamma _{ij}$ generate a pseudogroup $\Gamma$ of transformations on T (called the holonomy pseudogroup). The space of leaves $M/{\mathcal {F}}$ of the foliation ${\mathcal {F}}$ can be identified with $T/\Gamma$.

Definition 2.2

A smooth form $\omega$ on M is called basic if for any vector field X tangent to the leaves of ${\mathcal {F}}$ the following equality holds:

$$\begin{aligned} i_X\omega =i_Xd\omega =0. \end{aligned}$$

Basic 0-forms will be called basic functions henceforth.

Basic forms are in one to one correspondence with $\Gamma$-invariant smooth forms on T. It is clear that $d\omega$ is basic for any basic form $\omega$. Hence, the set of basic forms of ${\mathcal {F}}$ (denoted $\Omega ^{\bullet }(M/{\mathcal {F}})$) is a subcomplex of the de Rham complex of M. We define the basic cohomology of ${\mathcal {F}}$ to be the cohomology of this subcomplex and denote it by $H^{\bullet }(M/{\mathcal {F}})$. A transverse structure to ${\mathcal {F}}$ is a $\Gamma$-invariant structure on T. For example:

Definition 2.3

${\mathcal {F}}$ is said to be transversely symplectic if T admits a $\Gamma$-invariant closed 2-form $\omega$ of maximal rank. $\omega$ is then called a transverse symplectic form. As we noted earlier, $\omega$ corresponds to a closed basic form of rank q on M (also denoted $\omega$).

Definition 2.4

${\mathcal {F}}$ is said to be transversely holomorphic if T admits a complex structure that makes all the $\gamma _{ij}$ holomorphic. This is equivalent to the existence of an almost complex structure J on the normal bundle $N{\mathcal {F}}:=TM/T{\mathcal {F}}$ (where $T{\mathcal {F}}$ is the bundle tangent to the leaves) satisfying:

• ${\mathcal {L}}_XJ=0$ for any vector field X tangent to the leaves.

• if $Y_1$ and $Y_2$ are sections of the normal bundle then:

  $$\begin{aligned} N_J(Y_1,Y_2):=[JY_1,JY_2]-J[Y_1,JY_2]-J[JY_1,Y_2]+J^2[Y_1,Y_2]=0 \end{aligned}$$

  where [, ] is the bracket induced on the sections of the normal bundle (which can be defined by a choice of complement N of $T{\mathcal {F}}$ via $\pi _N([$, ])).

Remark 2.5

If ${\mathcal {F}}$ is transversely holomorphic we have the standard decomposition of the space of complex valued forms $\Omega ^{\bullet }({M/{\mathcal {F}},{\mathbb {C}}})$ into forms of type (p,q) and d decomposes into the sum of operators $\partial$ and ${\bar{\partial }}$ of order (1,0) and (0,1) respectively. Hence, one can define the Dolbeault double complex $(\Omega ^{\bullet ,\bullet }({M/{\mathcal {F}},{\mathbb {C}}}),\partial ,{\bar{\partial }})$, the Frölicher spectral sequence and the Dolbeault cohomology as in the manifold case.

Definition 2.6

${\mathcal {F}}$ is said to be transversely orientable if T is orientable and all the $\gamma _{ij}$ are orientation preserving. This is equivalent to the orientability of $N{\mathcal {F}}$.

Definition 2.7

${\mathcal {F}}$ is said to be Riemannian if T has a $\Gamma$-invariant Riemannian metric. This is equivalent to the existence of a Riemannian metric g on $N{\mathcal {F}}$ with $L_Xg=0$ for all vector fields X tangent to the leaves.

Definition 2.8

A foliation is said to be Hermitian if it is both transversely holomorphic and Riemannian.

Definition 2.9

A foliation ${\mathcal {F}}$ together with a triple $(g,J,\omega )$ consisting of a transverse Riemannian metric, transverse holomorphic structure and transverse symplectic form is called transversely Kähler if the following compatibility condition holds:

$$\begin{aligned} \omega (\cdot ,\cdot )=g(J\cdot ,\cdot )=\omega (J\cdot ,J\cdot ) \end{aligned}$$

We finish this section by recalling the spectral sequence of a Riemannian foliation.

Definition 2.10

We put:

$$\begin{aligned} F^k_{{\mathcal {F}}}\Omega ^r(M):=\{\alpha \in \Omega ^r(M)\text { }|\text { } i_{X_{r-k+1}}\cdots i_{X_1}\alpha =0,\text { for } X_1,\ldots ,X_{r-k+1}\in \Gamma (T{\mathcal {F}})\}. \end{aligned}$$

An element of $F^k_{{\mathcal {F}}}\Omega ^r(M)$ is called an r-differential form of filtration k.

The definition above in fact gives a filtration of the de Rham complex. Hence, via known theory from homological algebra we can cosntruct a spectral sequence as follows:

• (1) The 0-th page is given by $E_0^{p,q}=F_{{\mathcal {F}}}^{p}\Omega ^{p+q}(M)/F_{{\mathcal {F}}}^{p+1}\Omega ^{p+q}(M)$ and $d^{p,q}_0:E_0^{p,q}\rightarrow E^{p,q+1}_0$ is simply the morphism induced by d.

• (2) The r-th page is given inductively by:

  $$\begin{aligned} E_{r}^{p,q}:=Ker(d^{p,q}_{r-1})/Im(d^{p,q}_{r-1})=\frac{\{\alpha \in F^{p}_{{\mathcal {F}}}\Omega ^{p+q}(M)\text { }|\text { } d\alpha \in F_{{\mathcal {F}}}^{p+r}\Omega ^{p+q+1}(M)\}}{F^{p+1}_{{\mathcal {F}}}\Omega ^{p+q}(M) +d(F_{{\mathcal {F}}}^{p-r+1}\Omega ^{p+q-1}(M))} \end{aligned}$$

• (3) The r-th coboundary operator $d_r:E_r^{p,q}\rightarrow E^{p+r,q-r+1}_r$ is again just the map induced by d (due to the description of the r-th page this has the target specified above and is well defined).

Furthermore, since the filtration is bounded this spectral sequence converges and its final page is isomorphic to the cohomology of the cochain complex (in this case the de Rham cohomology of M).

Remark 2.11

The above spectral sequence can be thought of as a generalization of the Leray-Serre spectral sequence in de Rham cohomology to arbitrary Riemannian foliations (as opposed to fiber bundles).

2.2 Basic Hodge theory

We devote this section to provide some background information on basic Hodge theory (see [9]) which will be applied in the final two section of this article. Firstly, we recall a special class of Riemannian foliations on which the aforementioned theory is greatly simplified:

Definition 2.12

A codimension q foliation ${\mathcal {F}}$ on a connected manifold M is called homologically orientable if $H^{q}(M/{\mathcal {F}})={\mathbb {R}}$. A foliation ${\mathcal {F}}$ on a manifold M is called homologically orientable if its restriction to each connected component of M is homologically orientable.

We will see in the subsequent section that all the foliations considered in this paper are in fact homologically orientable and hence we shall restrict our attention to this case throughout the rest of this subsection.

Let ${\mathcal {F}}$ be a homologically orientable Riemannian foliation on a manifold M. One can use the transverse Riemannian metric to define the basic Hodge star operator $*_b$ pointwise. This in turn allows us to define the basic adjoint operator:

$$\begin{aligned} \delta _b=(-1)^{q(r+1)+1}*_bd*_b. \end{aligned}$$

Remark 2.13

While we choose this to be the definition of $\delta _b$, it is in fact an adjoint of d with respect to an appropriate inner product on forms induced by the transverse metric g. However, the definition of this inner product is quite involved and not necessary for our purpose. Although, we shall state some of the classical results of basic Hodge theory which use this inner product in their proof. For details see [9].

Using $\delta _b$ we can define the basic Laplace operator via:

$$\begin{aligned} \Delta _b=d\delta _b+\delta _bd. \end{aligned}$$

As it turns out this operator has some nice properties similar to that of the classical Laplace operator. In particular, it is transversely elliptic in the following sense:

Definition 2.14

A basic differential operator of order m is a linear map $D:\Omega ^{\bullet }(M/{\mathcal {F}})\rightarrow \Omega ^{\bullet }(M/{\mathcal {F}})$ such that in local coordinates $(x_1,\ldots ,x_p,y_1,\ldots ,y_q)$ (where $x_i$ are leaf-wise coordinates and $y_j$ are transverse ones) it has the form:

$$\begin{aligned} D=\sum \limits _{|s|\le m}a_s(y)\frac{\partial ^{|s|}}{\partial ^{s_1}y_1\cdots \partial ^{s_q}y_q} \end{aligned}$$

where $a_s$ are matrices of appropriate size with basic functions as coefficients. A basic differential operator is called transversely elliptic if its principal symbol is an isomorphism at all points of $x\in M$ and all non-zero, transverse, cotangent vectors at x.

In particular, this implies the following important result from [9]:

Theorem 2.15

Let ${\mathcal {F}}$ be a Riemannian homologically orientable foliation on a compact manifold M. Then:

• (1) $H^{\bullet }(M/{\mathcal {F}})$ is isomorphic to the space of basic harmonic forms $Ker(\Delta _b)$. In particular, it is finitely dimensional.

• (2) The basic Hodge star induces an isomorphism between $H^{k}(M/{\mathcal {F}})$ and $H^{q-k}(M/{\mathcal {F}})$ given by taking the class of the image through $*_b$ of a harmonic representative.

2.3 Primitive basic cohomology

Here we will recall some basic symplectic Hodge theory with main focus on primitive basic cohomology. To the best of our knowledge there is no concise source on the subject in its full generality (some special cases are treated in [6, 14]), hence for the readers convienience we provide a proof for the existence of basic primitive representatives and the Lefschetz decomposition on basic harmonic forms of transversely Kähler foliations.

Throughout this subsection let M be a compact manifold endowed with a homologically orientable, transversely symplectic Riemannian foliation ${\mathcal {F}}$ of codimension 2n. Firstly, let us note that by using the symplectic structure on the normal bundle $N{\mathcal {F}}$ we can define a symplectic star operator $*_s$ fiber by fiber in the standard way. This operator can in turn be used to define a number of other operators (on transverse forms i.e. sections of $\bigwedge ^{\bullet } N^*{\mathcal {F}}$ which can be naturally identified with differential forms satisfying $i_{X}\alpha =0$ for $X\in \Gamma (T{\mathcal {F}})$) of interest:

$$\begin{aligned} L\alpha :=\omega \alpha ,\quad \Lambda :=*_sL*_s,\quad d^{\Lambda }:=(-1)^{k+1}*_sd*_s=d\Lambda -\Lambda d. \end{aligned}$$

Remark 2.16

Note that, if $(g,\omega ,J)$ is a compatible triple consisting of a transverse Riemannian metric, transverse symplectic form and transverse almost complex structure, then by straightforward computation:

$$\begin{aligned} J*_s=*_b, \end{aligned}$$

where J acts on a k-form via:

$$\begin{aligned} (J\alpha )(X_1,\ldots ,X_k)=\alpha (JX_1,\ldots ,JX_k). \end{aligned}$$

this in turn allows one to compute that $\Lambda$ is dual to L with respect to the metric g.

We also denote by L the morphism induced in cohomology by L. We recall the solution to the basic Bryliński conjecture from [3]:

Theorem 2.17

Let M be a compact manifold endowed with a homologically orientable, transversely symplectic Riemannian foliation ${\mathcal {F}}$. Then the following conditions are equivalent:

• (1) (basic hard Lefschetz property) The map $L^k:H^{n-k}(M/{\mathcal {F}})\rightarrow H^{n+k}(M/{\mathcal {F}})$ is an isomorphism for all k.

• (2) Every basic cohomology class has a $d^{\Lambda }$-closed representative.

We move to some results on basic primitive forms required in this paper.

Definition 2.18

A basic $(n-k)$-form $\alpha$ is said to be primitive if $L^{k+1}\alpha =0$ (or equivalently $\Lambda \alpha =0$) for $k\in {\mathbb {N}}$. Similarly, a basic cohomology class $[\alpha ]$ of degree $(n-k)$ is said to be primitive if $L^{k+1}[\alpha ]=0$. The space of basic primitive cohomology classes is denoted by ${\mathcal {P}}H^{\bullet }(M/{\mathcal {F}})$.

The notion of primitive forms gives rise to the so called Lefschetz decomposition of forms:

Proposition 2.19

Let M be a compact manifold endowed with a homologically orientable, transversely symplectic Riemannian foliation ${\mathcal {F}}$ and let $\alpha$ be a basic r-form. Then $\alpha$ can be uniquely decomposed as:

$$\begin{aligned} \alpha =\sum \limits _{i} \omega ^i\beta _{r-2i}, \end{aligned}$$

where $\beta _{r-2i}$ are basic primitive forms of degree $r-2i$ and are given by the formula:

$$\begin{aligned} \beta _{r-2i}= \left( \sum _k\frac{a_{i,k}}{k!}L^k\Lambda ^{i+k}\alpha \right) , \end{aligned}$$

where $a_{i,k}$ are constants depending only on (n, i, k).

Proof

The proof of the unique decomposition and the explicit formula for $\beta _{r-2i}$ is well known from linear algebra (i.e. it is precisely the same as in the manifold case). The fact that the forms $\beta _{r-2i}$ are basic follows from the explicit formula for $\beta _{r-2i}$. $\square$

The following two theorems show that if a manifold satisfies the basic hard Lefschetz property then the above decomposition descends to cohomology.

Theorem 2.20

Let M be a compact manifold endowed with a homologically orientable, transversely symplectic Riemannian foliation ${\mathcal {F}}$ satisfying the basic hard Lefschetz property. Let $\alpha$ be a basic cohomology class of degree r. Then $\alpha$ can be uniquely decomposed as:

$$\begin{aligned} \alpha =\sum \limits _{i} \omega ^i\beta _{r-2i}, \end{aligned}$$

where $\beta _{r-2i}$ are basic primitive cohomology classes.

Proof

Firstly, let us note that by the basic hard Lefschetz property for $i<n$ the mapping $L:H^i(M/{\mathcal {F}})\rightarrow H^{i+2}(M/{\mathcal {F}})$ is a monomorphism. In particular, this means that for $r\le n$ we have:

$$\begin{aligned} H^r (M/{\mathcal {F}})={\mathcal {P}}H^r(M/{\mathcal {F}})\oplus LH^{r-2}(M/{\mathcal {F}}). \end{aligned}$$

Proceeding inductively we get:

$$\begin{aligned} H^r(M/{\mathcal {F}})=\bigoplus \limits _i L^i{\mathcal {P}}H^{r-2i}(M/{\mathcal {F}}). \end{aligned}$$

which proves the theorem for $r\le n$. For $r>n$ simply compose the above decomposition for $2n-r$ with $L^{r-n}$ and apply the hard Lefschetz property. $\square$

Theorem 2.21

Let M be a compact manifold endowed with a homologically orientable, transversely symplectic Riemannian foliation ${\mathcal {F}}$ satisfying the basic hard Lefschetz property. Every basic primitive cohomology class has a basic primitive representative.

Proof

Let $\alpha$ be a $d^{\Lambda }$-closed representative of a given basic primitive cohomology class. Then each primitive component of $\alpha$ is given by:

$$\begin{aligned} \beta _{r-2i}= \left( \sum _k\frac{a_{i,k}}{k!}L^k\Lambda ^{i+k}\alpha \right) , \end{aligned}$$

as described earlier. By applying d to the left hand side and noting that:

• (1) d commutes with L,

• (2) d commutes with $\Lambda$ up to $d^{\Lambda }$,

• (3) $d^{\Lambda }$ commutes with $\Lambda$,

we see that each $\beta _{r-2i}$ is closed. We note that each component aside from $\beta _r$ has to be also exact as otherwise $[\alpha ]$ would not be primitive. Hence, $[\beta _r]=[\alpha ]$ which ends the proof. $\square$

Finally, we give a similar decomposition theorem for basic harmonic forms on transversely Kähler foliations.

Theorem 2.22

Let M be a compact manifold endowed with a homologically orientable, transversely Kähler foliation ${\mathcal {F}}$. Let $\alpha$ be a basic harmonic r-form. Then $\alpha$ can be uniquely decomposed as:

$$\begin{aligned} \alpha =\sum \limits _{i} \omega ^i\beta _{r-2i}, \end{aligned}$$

where $\beta _{r-2i}$ are basic harmonic forms which are primitive.

Proof

We start by proving that if $\alpha$ is a basic harmonic k-form, then so is $L\alpha$. It is known (cf. [9]) that if a basic form on a transversely Kähler foliation is basic harmonic, then it is in the kernel of the operators $\partial$, ${\bar{\partial }}$ and their adjoints (with respect to the transverse metric). In particular, it is in the kernel of the adjoint $(d^c)^*$ of the operator:

$$\begin{aligned} d^c:= i({\bar{\partial }}-\partial )=J^{-1}dJ. \end{aligned}$$

Similarly as in the classical case we get:

$$\begin{aligned} d^{\Lambda }\alpha =(-1)^{k+1}*_sd*_s=(-1)^{k+1}*J^{-1}d*J^{-1}=(-1)^{k+1}*d^c*J^{-2}=(d^c)^{*}. \end{aligned}$$

hence we have proved that $0=d^{\Lambda }\alpha =d\Lambda \alpha$. This together with $\delta \Lambda \alpha =\Lambda \delta \alpha =0$ implies that $\Lambda \alpha$ is harmonic. By adjointess, the fact that $\Lambda$ preserves being harmonic implies that L does so as well.

Now if $\alpha$ is a basic harmonic form representing a primitive class then the form $L^{n-k+1}\alpha$ is harmonic as well. However, since $L^{n-k+1}\alpha$ represents the trivial cohomology class it has to be equal to 0. Hence, $\alpha$ is a primitive form itself. Now the theorem follows from Theorem 2.20 and the conclusion of the previous paragraph. $\square$

2.4 ${\mathcal {K}}$-structures and ${\mathcal {S}}$-structures

In this section we recall some of the work from [5] (see also [19, 20]). We start with some definitions:

Definition 2.23

An f-structure on a manifold $M^{2n+s}$ is a (1, 1) tensor field satisfying $f^3+f=0$.

As mentioned in the introduction the existence of such a structure is equivalent to a reduction of the structure group of M to $U(n)\times O(s)$. Throughout the paper we will use the notation “$M^{2n+s}$” to indicate that M is a $(2n+s)$-dimensional manifold endowed with an f-structure with $rank(f)=2n$.

Definition 2.24

We say that an f-structure on $M^{2n+s}$ has complemented frames if there are vector fields $\xi _i$ together with 1-forms $\eta _i$ for $i\in \{1,\ldots ,s\}$ satisfying:

$$\begin{aligned}{} & {} \eta _i(\xi _j)=\delta _{ij},\quad f\xi _i=0,\quad \eta _i\circ f=0,\\{} & {} \quad \quad f^2=-I+\sum _{k=1}^s\xi _k\otimes \eta _k \end{aligned}$$

for all $i,j\in \{1,\ldots ,s\}$.

Remark 2.25

We list some of the immediate properties of f-structures with complemented frames:

• (1) Ker(f) is equal to the bundle $<\xi _1,\ldots ,\xi _s>$ spanned pointwise by $\{\xi _1,\ldots ,\xi _s\}$. In particular this gives a partition $TM=Im(f)\oplus <\xi _1,\ldots ,\xi _s>$.

• (2) A complemented frame admits a compatible Riemannian metric g, i.e. such that:

  $$\begin{aligned} g(X,Y)=g(fX,fY)+\sum _{k=1}^s \eta _k(X)\eta _k(Y). \end{aligned}$$

  with respect to such a metric the forms $\eta _i$ are dual to the corresponding vector fields $\xi _i$.

• (3) A compatible metric allows us to define a 2-form $F(\bullet ,\bullet ):=g(\bullet ,f\bullet )$ which is non-degenerate on Im(f). It is easy to see that $\eta _1\cdots \eta _s F^n\ne 0$. This implies that M is orientable. Throughout the rest of the paper we will consider M with the orientation induced by the above $(2n+s)$-form

We now specialize to ${\mathcal {K}}$-manifolds introduced in [5]. However, our main results can be applied to a slightly more general class of manifolds which is more natural to define beforehand:

Definition 2.26

A manifold $M^{2n+s}$ together with an f-structure with a chosen complemented frame $\{\xi _1,\ldots ,\xi _s,\eta _1,\ldots ,\eta _s\}$ and a compatible Riemannian metric g is an almost ${\mathcal {K}}$-manifold if:

• (1) The 2-form $F(\bullet ,\bullet ):=g(\bullet ,f\bullet )$ is closed.

• (2) The vector fields $\{\xi _1,\ldots ,\xi _s\}$ are Killing.

If in addition the above set of data satisfies the equation:

$$\begin{aligned}{}[f,f]+\sum _{k=1}^{s} \xi _k\otimes d\eta _k=0, \end{aligned}$$

where [f, f] is the Nijenhuis tensor of f, then the almost ${\mathcal {K}}$-structure is said to be integrable. An integrable almost ${\mathcal {K}}$-structure is also called a ${\mathcal {K}}$-structure (cf. [5]).

Proposition 2.27

Let $M^{2n+s}$ be an almost ${\mathcal {K}}$-manifold. Then Ker(f) is involutive and hence induces a foliation.

Proof

It follows from the definition that Ker(f) is equal to the kernel of the closed 2-form $F(X,Y):=g(X,f Y)$ (treated as a map from TM to $T^*M$). Hence, for any vector field X we get the following equalities:

$$\begin{aligned} 0=dF(\xi _i,\xi _j,X)= & {} {\mathcal {L}}_{\xi _i}(F(\xi _j,X))-{\mathcal {L}}_{\xi _j}(F(\xi _i,X)) +{\mathcal {L}}_{X}(F(\xi _i,\xi _j)) \\ {}{} & {} -F([\xi _i,\xi _j], X)+F([\xi _i,X], \xi _j)-F([\xi _j,X], \xi _i) \\ {}= & {} -F([\xi _i,\xi _j], X). \end{aligned}$$

Which means that $[\xi _i,\xi _j]$ is again in the kernel of F and hence the involutivity of Ker(f) follows. $\square$

Moreover, it is clear that the foliation above is Riemannian (with $g(f\bullet ,f\bullet )$) and transversely symplectic (with the 2-form F), while the integrability condition implies that the foliation is also transversely holomorphic (and hence transversely Kähler). An almost ${\mathcal {K}}$-structure not only defines the above foliation with its additional structures but also connects the transverse geometry of the foliation to that of the entire manifold. One instance of this is the following proposition which we will use later on in some of our applications.

Proposition 2.28

Let $M^{2n+s}$ be an almost ${\mathcal {K}}$-manifold and let $i=(i_1,\ldots ,i_k)$ be an ordered subset of $\{1,\ldots ,s\}$ with complement $j=(j_1,\ldots ,j_{s-k})$. Then the following relation between the hodge star operator $*$ and the basic hodge star operator $*_{b}$ holds for any transverse r-form $\alpha$ (i.e. $i_{\xi _l}\alpha =0$):

$$\begin{aligned} *(\eta _{i_1}\cdots \eta _{i_k}\alpha )=(-1)^{sign(i_1,\ldots ,i_k,j_1,\ldots ,j_{s-k}) +(s-k)r}\eta _{j_1}\cdots \eta _{j_{s-k}}*_b\alpha \end{aligned}$$

Among ${\mathcal {K}}$-structure the notions of ${\mathcal {S}}$-structure and ${\mathcal {C}}$-structures were given special attention in [5] due to them being the proper generalization of Sasakian and quasi-Sasakian with $d\eta =0$ cases to the above setting and as such exhibit analogous curvature properties. Again we introduce these structures proceeded by their ”almost structure” counterpart:

Definition 2.29

An almost ${\mathcal {K}}$-structure on a manifold $M^{2n+s}$:

• (1) Is called an almost ${\mathcal {S}}$-structure if $d\eta _i=F$ for all $i\in \{1,\ldots ,s\}$.

• (2) Is called an almost ${\mathcal {C}}$-structure if $d\eta _i=0$ for all $i\in \{1,\ldots ,s\}$.

Moreover, if the underlying almost ${\mathcal {K}}$-structure of an almost ${\mathcal {S}}$-structure (resp. almost ${\mathcal {C}}$-structure) is integrable, then it is called a ${\mathcal {S}}$-structure (resp. ${\mathcal {C}}$-structure).

We finish this section by reiterating for the readers convenience the analogy between the above structures and their low dimensional counterparts.

s = 0	s = 1	General
–	Quasi-K-contact	Almost ${\mathcal {K}}$
–	Quasi-Sasaki	${\mathcal {K}}$
Almost Kähler	K-contact	Almost ${\mathcal {S}}$
Kähler	Sasaki	${\mathcal {S}}$
–	Quasi-K-contact with $d\eta =0$	Almost ${\mathcal {C}}$
–	Quasi-Sasaki with $d\eta =0$	${\mathcal {C}}$

Remark 2.30

It is also worth noting that our almost ${\mathcal {S}}$-manifolds are already present in the literature as f-K-contact manifolds. However, we choose to stick to our terminology as it seems more appropriate when considering such manifolds along with non-integrable versions of ${\mathcal {K}}$-manifolds and ${\mathcal {C}}$-manifolds.

3 The spectral sequence of invariant forms on almost ${\mathcal {K}}$-manifolds

Here we describe a canonical torus action on certain almost ${\mathcal {K}}$-manifolds and use it to define a spectral sequence used in further chapters. We start with the following proposition:

Proposition 3.1

Let $M^{2n+s}$ be an almost ${\mathcal {K}}$-manifold, such that for each $l\in \{1,\ldots ,s\}$ the form $d\eta _l$ is basic. Then for each $i,j\in \{1,\ldots ,s\}$ the equality $[\xi _i,\xi _j]=0$ holds.

Proof

This follows from the computation:

$$\begin{aligned} 0=d\eta _l(\xi _i,\xi _j)={\mathcal {L}}_{\xi _i}\eta _l(\xi _j) -{\mathcal {L}}_{\xi _j}\eta _l(\xi _i)-\eta _{l}([\xi _i,\xi _j])=-\eta _l([\xi _i,\xi _j]). \end{aligned}$$

Now since Ker(f) is involutive, it follows that if $[\xi _i,\xi _j]\ne 0$ then there exists some l such that $\eta _l([\xi _i,\xi _j])\ne 0$ which provides the desired contradiction with the computation above. $\square$

Remark 3.2

Let us note that in particular almost ${\mathcal {S}}$-manifolds and almost ${\mathcal {C}}$-manifolds satisfy the assumptions of the above proposition. Hence, this proposition as well as the remainder of this section can be applied to them.

This has the following important corollary:

Corollary 3.3

Let $M^{2n+s}$ be a compact almost ${\mathcal {K}}$-manifold, such that for each $i\in \{1,\ldots ,s\}$ the form $d\eta _i$ is basic. Let $G\subset Diff(M)$ be the group whose Lie algebra is $<\xi _1,\ldots ,\xi _s>\subset \Gamma (TM)$. Then the closure of G is a torus in the group Isom(M) of isometries on M.

Proof

Let us first note that since the vector fields $\xi _i$ are Killing we have the inclusion $G\subset Isom(M)$ which is known to be a finitely dimensional compact Lie group. Moreover, since by Proposition 3.1G is abelian its closure is a compact abelian group and hence a torus. $\square$

The next step is to classify forms on M which are invariant under the action of ${\overline{G}}$.

Proposition 3.4

Let $M^{2n+s}$ be a compact almost ${\mathcal {K}}$-manifold such that for each $i\in \{1,\ldots ,s\}$ the form $d\eta _i$ is basic. Then the following conditions are equivalent:

• (1) $\alpha$ is a ${\overline{G}}$-invariant form on M.

• (2) $\alpha =\alpha _0+\sum \limits _{k=1}^s\sum \limits _{1\le i_1<\cdots <i_k\le s}\eta _{i_1}\cdots \eta _{i_k}\alpha _{i_1,\ldots ,i_k}$, where $\alpha _0$ and $\alpha _{i_1,\ldots ,i_k}$ are basic for all indices $1\le i_1<\cdots <i_k\le s$.

Proof

Assume that the second condition is true. Then it can be easilly computed that for any $\xi _j$ the equality ${\mathcal {L}}_{\xi _j}\alpha =0$ holds. Which in turn implies that $\alpha$ is ${\overline{G}}$-invariant.

Now let us write the invariant form $\alpha$ as:

$$\begin{aligned} \alpha =\alpha _0+\sum \limits _{k=1}^s\sum \limits _{1\le i_1<\cdots , i_k\le s}\eta _{i_1}\cdots \eta _{i_k}\alpha _{i_1,\ldots ,i_k}, \end{aligned}$$

where $\alpha _{i_1,\ldots ,i_k}$ are transverse for all indices $1\le i_1<\cdots <i_k\le s$. Due to the well known formula:

$$\begin{aligned} {\mathcal {L}}_{X}i_{Y}-i_{Y}{\mathcal {L}}_{X}=i_{[X,Y]}, \end{aligned}$$

we get that $i_{\xi _i}$ and ${\mathcal {L}}_{\xi _j}$ commute for $i,j\in \{1,\ldots ,s\}$ (using Proposition 3.1). We shall now prove that the forms $\alpha _0$ and $\alpha _{i_1,\ldots ,i_k}$ are basic by reverse induction on the number of indices. Hence, we start by proving that $\alpha _{1,\ldots ,s}$ is basic. Since $\alpha$ is harmonic and the vector fields $\xi _i$ are Killing we have for any $i\in \{1,\ldots ,s\}$ the following equalities:

$$\begin{aligned} 0={\mathcal {L}}_{\xi _i}\alpha =i_{\xi _s}i_{\xi _{s-1}}\cdots i_{\xi _1}{\mathcal {L}}_{\xi _i}\alpha ={\mathcal {L}}_{\xi _i}i_{\xi _{s}}i_{\xi _{s-1}}\cdots i_{\xi _1}\alpha ={\mathcal {L}}_{\xi _i}\alpha _{1,\ldots ,s}. \end{aligned}$$

which proves that $\alpha _{1,\ldots ,s}$ is basic.

For the induction step let us assume that all the $\alpha _{i_1,\ldots ,i_{k}}$ for $s\ge k>K$ are basic. We shall show that all $\alpha _{i_1,\ldots ,i_{K}}$ are basic as well. Using the assumption we get for any $i\in \{1,\ldots ,s\}$ the following equalities:

$$\begin{aligned} 0={\mathcal {L}}_{\xi _i}\alpha =i_{\xi _{i_K}}i_{\xi _{i_{K-1}}}\cdots i_{\xi _{i_1}}{\mathcal {L}}_{\xi _i}\alpha ={\mathcal {L}}_{\xi _i}i_{\xi _{i_K}}i_{\xi _{i_{K-1}}}\cdots i_{\xi _{i_1}}\alpha ={\mathcal {L}}_{\xi _i}\alpha _{i_1,\ldots ,i_K}. \end{aligned}$$

Which proves that $\alpha _{i_1,\ldots ,i_K}$ are basic for any set of indices $1\le i_1<\cdots <i_k\le s$. $\square$

Remark 3.5

Note that the induction assumption is used to pass to the final equality as it implies that all the terms with a greater number of indices then K vanish under ${\mathcal {L}}_{\xi _i}$ as:

$$\begin{aligned} {\mathcal {L}}_{\xi _i}\eta _{j_1}\cdots \eta _{j_k}\alpha _{j_1,\ldots ,j_k} =\eta _{j_1}\cdots \eta _{j_k}{\mathcal {L}}_{\xi _i}\alpha _{j_1,\ldots ,j_k}=0. \end{aligned}$$

The first equality is due to the fact that ${\mathcal {L}}_{\xi _i}\eta _j=i_{\xi _i}d\eta _j+d(i_{\xi _i}\eta _j)=0$.

Finally, we note that similarly as for the spectral sequence of a Riemannian foliation we have a filtration of the cochain complex of invariant forms $\Omega ^r_{{\overline{G}}}(M)$ given by:

$$\begin{aligned} F^k_{{\mathcal {F}}}\Omega ^r_{{\overline{G}}}(M):=\{\alpha \in \Omega ^r_{{\overline{G}}}(M)\text { }|\text { } i_{X_{r-k+1}}\cdots i_{X_1}\alpha =0,\text { for } X_1,\ldots ,X_{r-k+1}\in \Gamma (T{\mathcal {F}})\}. \end{aligned}$$

Hence, via known theory from homological algebra we can cosntruct a spectral sequence as follows:

• (1) The 0-th page is given by $E_0^{p,q}=F_{{\mathcal {F}}}^{p}\Omega ^{p+q}_{{\overline{G}}}(M)/F_{{\mathcal {F}}}^{p+1}\Omega ^{p+q}_{{\overline{G}}}(M)$ and $d^{p,q}_0:E_0^{p,q}\rightarrow E^{p,q+1}_0$ is simply the morphism induced by d.

• (2) The r-th page is given inductively by:

  $$\begin{aligned} E_{r}^{p,q}:=Ker(d^{p,q}_{r-1})/Im(d^{p,q}_{r-1})=\frac{\{\alpha \in F^{p}_{{\mathcal {F}}}\Omega ^{p+q}_{{\overline{G}}}(M)\text { }|\text { } d\alpha \in F_{{\mathcal {F}}}^{p+r}\Omega ^{p+q+1}_{overline{G}}(M)\}}{F^{p+1}_{{\mathcal {F}}} \Omega ^{p+q}_{{\overline{G}}}(M)+d(F_{{\mathcal {F}}}^{p-r+1}\Omega _{{\overline{G}}}^{p+q-1}(M))} \end{aligned}$$

• (3) The r-th coboundary operator $d_r:E_r^{p,q}\rightarrow E^{p+r,q-r+1}_r$ is again just the map induced by d (due to the description of the r-th page this has the target specified above and is well defined).

Furthermore, since the filtration is bounded this spectral sequence converges and its final page is isomorphic to the cohomology of the cochain complex $\Omega ^{r}_{{\overline{G}}}(M)$ known to be isomorphic to the de Rham cohomology of M. We call this spectral sequence the spectral sequence of invariant forms and denote it by $E_{r}^{p,q}$ throughout the rest of the paper.

Theorem 3.6

Let $M^{2n+s}$ be a compact almost ${\mathcal {S}}$-manifold such that for each $i\in \{1,\ldots ,s\}$ the form $d\eta _i$ is basic. Then:

$$\begin{aligned} E_2^{p,q}\cong \bigwedge \text {}_{H^{p}(M/{\mathcal {F}})}^q<\eta _1,\ldots ,\eta _s>:=H^{p}(M/{\mathcal {F}})\otimes \bigwedge \text {} ^q<\eta _1,\ldots ,\eta _s>. \end{aligned}$$

Proof

Since the operator d takes basic forms to basic forms and $d\eta _i$ is basic for all $i\in \{1,\ldots ,s\}$ it is easy to see that $d_0$ is in fact equal to the zero operator. Hence, the first page is isomorphic to the 0-th page.

On the first page by the same observation the operator $d_1$ is just the application of d to the transverse part of the form (since applying d to $\bigwedge ^q<\eta _1,\ldots ,\eta _s>$ decrease q). Hence, the second page is just $H^{p}(M/{\mathcal {F}})\otimes \bigwedge ^q<\eta _1,\ldots ,\eta _s>$. $\square$

Remark 3.7

• (1) We note that the merit of considering invariant forms is already visible in this computation since a similar result for the spectral sequence of the Riemannian foliation is not known. In fact, it is far from trivial to even prove that the second page of this spectral sequence is finitely dimensional (cf. [1]). On a more down to earth level the major simplification comes from the triviality of $d_0$ in the spectral sequence of invariant forms.

• (2) It is interesting to note that for ${\mathcal {S}}$-manifolds the above description of $E^{p,q}_2$ coincides (disregarding the coboundary operators) with the "almost formal" models from [7].

• (3) For the sake of brevity the notation:

  $$\begin{aligned} \bigwedge \text {}_{V}^q<\eta _1,\ldots ,\eta _s>:=V\otimes \bigwedge \text {} ^q<\eta _1,\ldots ,\eta _s>, \end{aligned}$$

  introduced in the above theorem, shall be used throughout the article. We shall also use its following variations:

  $$\begin{aligned} \bigwedge \text {}_{V}^{\bullet }<\eta _1,\ldots ,\eta _s>:=V\otimes \bigwedge \text {} ^{\bullet }<\eta _1,\ldots ,\eta _s>, \end{aligned}$$
  $$\begin{aligned} {\overline{\bigwedge }}\text {}_{V}^{\bullet }<\eta _1,\ldots ,\eta _s>:=\{\alpha \in V\otimes \bigwedge \text {} ^{\bullet }<\eta _1,\ldots ,\eta _s>\text { }|\text { } \pi _{V\otimes 1}\alpha =0\}, \end{aligned}$$

  where $\pi _{V\otimes 1}$ is the obvious projection onto $V\otimes 1\subset \bigwedge \text {}_{V}^{\bullet }<\eta _1,\ldots ,\eta _s>.$

We also wish to mention the following consequence of the above discussion which will be used throughout the paper in order to omit the homological orientability assumption throughout the rest of the article:

Proposition 3.8

Let $M^{2n+s}$ be a compact almost ${\mathcal {S}}$-manifold such that for each $i\in \{1,\ldots ,s\}$ the form $d\eta _i$ is basic. Then the foliation induced by Ker(f) is homologically orientable.

Proof

It is well known (cf. [9]) that the top basic cohomology of a Riemannian foliation on a compact manifold is either 0 or ${\mathbb {R}}$. In this case it cannot be 0 since then we could compute from the above spectral sequence that $H^{2n+s}_{dR}(M)\cong E^{2n,s}_2=0$ which is a contradiction with the orientability of M. Hence, the top basic cohomology is isomorphic to ${\mathbb {R}}$ which means that the foliation is homologically orientable. $\square$

4 Main results for almost ${\mathcal {S}}$-manifolds

In this section we prove our main results for compact almost ${\mathcal {S}}$-manifolds. We start by computing $E^{p,q}_3$ for almost ${\mathcal {S}}$-manifolds satisfying the transverse hard Lefschetz property. Firstly, we compute the kernel of $d_2$.

Lemma 4.1

Under the above assumptions we have:

$$\begin{aligned} Ker(d_2)= \left( \bigwedge \text {}_{H^{\bullet }(M/{\mathcal {F}})}^{\bullet }<\eta _1-\eta _{2},\ldots ,\eta _1-\eta _{s}> +{\overline{\bigwedge }}\text {}_{Ker(L)}^{\bullet } <\eta _{1},\ldots ,\eta _{s}> \right) . \end{aligned}$$

Proof

Firstly, let us note that $Ker(d^{p,0}_2)$ is simply $H^{p}(M/{\mathcal {F}})$. For $q>0$, by Theorem 3.6 we can write an element $[\alpha ]$ of $E_2^{p,q}$ as:

$$\begin{aligned}{}[\alpha ]=\sum \limits _{1\le i_1<\cdots <i_q\le s}\eta _{i_1}\cdots \eta _{i_q}[\alpha _{i_1,\ldots ,i_q}], \end{aligned}$$

for some basic forms $[\alpha _{i_1,\ldots ,i_q}]$. Applying $d_2$ to this element gives:

$$\begin{aligned} d_2[\alpha ]=\sum \limits _{1\le i_1<\cdots <i_{q-1}\le s}\eta _{i_1}\ldots \eta _{i_{q-1}}L \left( \sum \limits _{j=1}^s[\alpha _{i_1,\ldots ,i_{q-1},j}] \right) \end{aligned}$$

where we understand $\alpha _{i_1,\ldots ,i_{q-1},j}$ to be equal to zero if $j\in \{i_1,\ldots ,i_{q-1}\}$ and to be equal to $sign(i_1,\ldots ,i_{q-1},j)\alpha _{i_1,\ldots ,j,\ldots ,i_{q-1}}$ otherwise (here j is on the correct position so that the indices are in an increasing order). This implies that ${\overline{\bigwedge }}_{Ker(L)}^{\bullet } <\eta _{1},\ldots ,\eta _{s}>$ is in fact contained in $Ker(d^{p,q}_2)$.

Here we split the consideration into two cases $p< n$ and $p\ge n$.

For the first case, L is a monomorphism and the elements $\sum \limits _{j=1}^s[\alpha _{i_1,\ldots ,i_{q-1},j}]$ have to be trivial for $\alpha$ to be an element of $Ker(d_2^{p,q})$. However, by assigning $\eta _{i_1}\ldots \eta _{i_k}$ to the simplex $[i_1,\ldots ,i_k]$ we get a commutative diagram:

The horizontal arrows in this diagram are isomorphisms and hence they induce an isomorphism on the kernels of the vertical arrows. But $Ker(\partial ^q)=Im(\partial ^{q+1})$. While for $Im(\partial ^{q+1})$ we can easilly determine the generators as the images of the simplices generating ${\tilde{C}}^{q+1}$. These in turn correspond to the elements of the form $(\eta _{i_1}-\eta _{i_2})\cdots (\eta _{i_1}-\eta _{i_q})\alpha$ for $\alpha \in H^{p}(M/{\mathcal {F}})$.

For the second case, Theorem 2.20 implies that, Ker(L) is equal to $L^{p-n}{\mathcal {P}}H^{2n-p}(M/{\mathcal {F}})$. Hence, in what follows it suffices to consider classes from $L^{p-n+1}H^{2n-p-2}(M/{\mathcal {F}})$ on which L is monomorphic. From here the same argument as in the first case can be conducted with the coefficients changed to $L^{p-n+1}H^{2n-p-2}(M/{\mathcal {F}})$. $\square$

With this we are ready to prove our main result concerning almost ${\mathcal {K}}$-manifolds:

Theorem 4.2

Let $M^{2n+s}$ be a compact almost ${\mathcal {K}}$-manifold satisfying the transverse hard Lefschetz property. Then:

$$\begin{aligned} H^{\bullet }_{dR}(M) \cong \bigwedge \text {}^{\bullet }_{{\mathcal {P}}H^{\bullet }(M/{\mathcal {F}})}<\eta _1-\eta _2,\ldots ,\eta _1 -\eta _s>\oplus \eta _1\bigwedge \text {}^{\bullet }_{Ker(L)}<\eta _2,\ldots ,\eta _s>. \end{aligned}$$

Proof

Firstly, let us note that the image of $d^{p,1}_2$ is equal to the image of L and hence $E^{p,0}_3\cong {\mathcal {P}}H^{p}(M/{\mathcal {F}})$.

Secondly, for $p<n$ we again have that L is a monomorphism and hence we have the identification:

This implies (similarly as in the previous proof) that the image consists of elements of the form $(\eta _{i_1}-\eta _{i_2})\cdots (\eta _{i_1}-\eta _{i_q})L\alpha$ for $\alpha \in H^{p-2}(M{\mathcal {F}})$. With this (together with Lemma 4.1) we conclude that in this range of p we have:

$$\begin{aligned} E_3^{p,q}\cong \bigwedge \text {}^{\bullet }_{{\mathcal {P}}H^{\bullet }(M/{\mathcal {F}})} <\eta _1-\eta _2,\ldots ,\eta _1-\eta _s>. \end{aligned}$$

Thirdly, we treat the case $p\ge n$. Here it is crucial to note that due to its form $d^{p,q}_2$ preserves the basic Lefschetz decomposition of each element of $E^{p,q}_2$. In particular, this allows us to consider seperately $d^{p-2,q+1}_2$ on $L^{p-n+2}H^{2n-p-4}(M/{\mathcal {F}})$ and $L^{p-n+1}{\mathcal {P}}H^{2n-p-2}(M/{\mathcal {F}})$.

For $L^{p-n+2}H^{2n-p-4}(M/{\mathcal {F}})$, a similar diagram as in the previous case (but with coefficients in $L^{p-n+2}H^{2n-p-4}(M/{\mathcal {F}})$) will allow us to compute the image of $d^{p-2,q+1}_2$ to consists of elements of the form $(\eta _{i_1}-\eta _{i_2})\cdots (\eta _{i_1}-\eta _{i_q})L\alpha$ for $\alpha \in L^{p-n+2}H^{2n-p-4}(M/{\mathcal {F}})$. But since in the given range of p the morphism L is an epimorphism this indicates that for such p the algebra $\bigwedge \text {}_{H^{\bullet }(M/{\mathcal {F}})/Ker(L)}^{\bullet } <\eta _1-\eta _{2},\ldots ,\eta _1-\eta _{s}>$ trivializes on $E^{p,q}_3$.

For $L^{p-n+1}{\mathcal {P}}H^{2n-p-2}(M/{\mathcal {F}})$, we again use the same method to compute that the image of $d^{p-2,q+1}_2$ consist of elements of the form $(\eta _{i_1}-\eta _{i_2})\cdots (\eta _{i_1}-\eta _{i_q})L\alpha$ for $\alpha \in L^{p-n+1}{\mathcal {P}}H^{2n-p-2}(M/{\mathcal {F}})$. This by the basic Lefschetz decomposition can be written alternatively as $(\eta _{i_1}-\eta _{i_2})\cdots (\eta _{i_1}-\eta _{i_q})\alpha$ for $\alpha \in Ker(L)$. Combining this with the previous paragraph we get that in this range of p the following holds:

$$\begin{aligned} E^{p,q}_3\cong {\overline{\bigwedge }}\text {}_{Ker(L)}^{\bullet }<\eta _{1},\ldots ,\eta _{s}>/{\overline{\bigwedge }}\text {}_{Ker(L)}^{\bullet } <\eta _1-\eta _{2},\ldots ,\eta _1-\eta _{s}>. \end{aligned}$$

This in turn can be easilly computed to be isomorphic to the complement of ${\overline{\bigwedge }}\text {}_{Ker(L)}^{\bullet } <\eta _1-\eta _{2},\ldots ,\eta _1-\eta _{s}>$ given for example by $\eta _1{\overline{\bigwedge }}\text {}^{\bullet }_{Ker(L)}<\eta _2,\ldots ,\eta _s>.$

Finally, we note that by the basic Lefschetz decomposition we can take the basic components of the representatives of the classes from $E_3^{p,q}$ to be either primitive closed forms or closed forms from Ker(L) which by the construction of the spectral sequence proves that the spectral sequence degenerates at the 3rd page. Moreover, by pinpointing the representatives in such a way we can conclude that they also represent the cohomology classes in $H^{\bullet }_{dR}(M)$. $\square$

Remark 4.3

We note that a different proof of Theorem 4.2 can be conducted by combining some recent results from [11] with methods from [14]. More precisely, Proposition 4.4 from [11] gives a Gysin-like long exact sequence connecting the basic cohomology of the foliation ${\mathcal {F}}_{s-1}$ spaned by $\{\xi _2,\ldots ,\xi _{s}\}$ and the basic cohomology of ${\mathcal {F}}$. By analyzing it (with the use of the basic hard Lefschetz property) similarly as in [14] we get:

$$\begin{aligned} H^{\bullet }(M/{\mathcal {F}}_{s-1})\cong {\mathcal {P}}H^{\bullet }\oplus \eta _1 Ker(L). \end{aligned}$$

By Theorem 4.5 from [11], which relates $H^{\bullet }(M/{\mathcal {F}}_{s-1})$ to $H^{\bullet }_{dR}(M)$ one arrives at the conclusion of Theorem 4.2.

An immediate consequence of Theorem 4.2 is the following important result on topological invariance of basic cohomology.

Corollary 4.4

Let $M^{2n+s}_1$ and $M^{2n+s}_2$ be compact almost ${\mathcal {S}}$-manifolds satisfying the hard Lefschetz property which are homeomorphic. Then the basic cohomologies of the corresponding foliations are isomorphic.

Proof

The above description can be used to compute the primitive basic cohomology of both the foliations ${\mathcal {F}}_l$ on $M_l$ for $l\in \{1,2\}$. More precisely it can be done inductively by the formula:

$$\begin{aligned} dim({\mathcal {P}}H^{k}(M_l/{\mathcal {F}}_l))=dim(H^{k}_{dR}(M_l))-\sum \limits ^{k-1}_{i=0} {s-1\atopwithdelims ()k-i} dim({\mathcal {P}}H^i(M_l/{\mathcal {F}}_l)) \end{aligned}$$

with the convention that ${s-1\atopwithdelims ()k-i}=0$ if $k-i>s-1$. This implies that:

$$\begin{aligned} dim({\mathcal {P}}H^{k}(M_1/{\mathcal {F}}_1)) =dim({\mathcal {P}}H^{k}(M_2/{\mathcal {F}}_2)), \end{aligned}$$

for all $0\le k\le n$. But these dimensions are enough to compute the dimensions of the basic cohomology of ${\mathcal {F}}$ using the Lefschetz decomposition. Hence:

$$\begin{aligned} dim(H^{k}(M_1/{\mathcal {F}}_1))=dim(H^{k}(M_2/{\mathcal {F}}_2)), \end{aligned}$$

for all $k\in {\mathbb {N}}$ which in turn implies that the basic cohomologies of ${\mathcal {F}}_1$ and ${\mathcal {F}}_2$ are isomorphic. $\square$

Remark 4.5

Unfortunately, similarly as in the Sasakian (and K-contact) case this method does not produce any cannonical isomorphism between the basic cohomologies of ${\mathcal {F}}_1$ and ${\mathcal {F}}_2$.

5 Main result for almost ${\mathcal {C}}$-manifolds

Theorem 5.1

Let $M^{2n+s}$ be an almost ${\mathcal {C}}$-manifold. Then:

$$\begin{aligned} H^{k}_{dR}(M)\cong \bigoplus \limits _{p+q=k} H^{p}(M/{\mathcal {F}})\otimes \bigwedge \text {} ^q<\eta _1,\ldots ,\eta _s>. \end{aligned}$$

Proof

In this case the operator d itself goes from $\Omega ^{p}(M/{\mathcal {F}})\otimes \bigwedge ^q<\eta _1,\ldots ,\eta _s>$ to $\Omega ^{p+1}(M/{\mathcal {F}})\otimes \bigwedge ^q<\eta _1,\ldots ,\eta _s>$. Hence, the spectral sequence degenerates at the second page which together with Theorem 3.6 implies the thesis. $\square$

It is worth noting that in this case the transverse hard Lefschetz property is not needed. In particular, a similar result holds for quasi-Sasakian manifolds with $d\eta =0$. Similarly as in the ${\mathcal {S}}$-manifold case this also implies that the basic cohomology of almost ${\mathcal {C}}$-manifolds are a topological invariant.

Corollary 5.2

Let $M^{2n+s}_1$ and $M^{2n+s}_2$ be compact almost ${\mathcal {C}}$-manifolds which are homeomorphic. Then the basic cohomologies of the induced foliations are isomorphic.

Proof

The above description can be used to compute the basic cohomology of both the foliations ${\mathcal {F}}_l$ on $M_l$ for $l\in \{1,2\}$. More precisely it can be done inductively by the formula:

$$\begin{aligned} dim(H^{k}(M_l/{\mathcal {F}}_l))=dim(H^{k}_{dR}(M)_l)-\sum \limits ^{k-1}_{i=0} {s\atopwithdelims ()k-i} dim(H^i(M_l/{\mathcal {F}}_l)) \end{aligned}$$

with the convention that ${s\atopwithdelims ()k-i}=0$ if $k-i>s$. This implies that:

$$\begin{aligned} dim(H^{k}(M_1/{\mathcal {F}}_1))=dim(H^{k}(M_2/{\mathcal {F}}_2)), \end{aligned}$$

for all $k\in {\mathbb {N}}$ which in turn implies that the basic cohomologies of ${\mathcal {F}}_1$ and ${\mathcal {F}}_2$ are isomorphic. $\square$

Remark 5.3

As with the analogous result from the previous chapter this method does not produce any cannonical isomorphism between the basic cohomologies of ${\mathcal {F}}_1$ and ${\mathcal {F}}_2$.

Remark 5.4

While it seems doubtful that similar general results can be achieved for arbitrary almost ${\mathcal {K}}$-manifolds we feel that the above spectral sequence remains a good tool to find similar dependencies in cohomology on a case by case basis.

6 Application: classification of harmonic forms

Here we provide a description of harmonic forms on almost ${\mathcal {C}}$-manifolds and ${\mathcal {S}}$-manifolds based on their basic harmonic forms. Firstly, let us note the following:

Remark 6.1

• (1) Theorem 4.2 allows us to treat $\bigwedge \text {}^{\bullet }_{{\mathcal {P}}H^{\bullet }(M/{\mathcal {F}})}<[\eta _1-\eta _2],\ldots ,[\eta _1-\eta _s]>$ and $\eta _1\bigwedge \text {}^{\bullet }_{Ker(L)}<\eta _2,\ldots ,\eta _s>$ as submodules of $H^{\bullet }_{dR}(M)$ with representatives given respectively by linear combinations of elements of the form $(\eta _{1}-\eta _{i_1})\cdots (\eta _1-\eta _{i_q})\alpha$ (where $\alpha$ is a closed basic primitive form) and $\eta _1\eta _{i_1}\cdots \eta _{i_s}\alpha$ (where $\alpha \in Ker(L)$ is a closed basic form).

• (2) Theorem 5.1 allows us to treat $[\eta _{i_1}]\cdots [\eta _{i_q}]H^{p}(M/{\mathcal {F}})$ as a submodule of $H^{p+q}_{dR}(M)$ with representatives given by linear combinations of elements of the form $\eta _{i_1}\cdots \eta _{i_q}\alpha$ (where $\alpha$ is a closed basic form).

Making the identification from the previous remark the following statements can be now made:

Theorem 6.2

Let $M^{2n+s}$ be a compact almost ${\mathcal {C}}$-manifold. A form $\alpha$ on M is harmonic if and only if it is a linear combination of elements of the form $\eta _{i_1}\cdots \eta _{i_q}{\tilde{\alpha }}$ such that ${\tilde{\alpha }}$ is basic harmonic.

Proof

We start by noting that any element $\alpha$ of the given form is in fact harmonic (which immediately implies that the linear combination of such elements is also harmonic), since through straightforward computation (with the use of Proposition 2.28) we can get $d\alpha =\delta \alpha =0$.

On the other hand given any harmonic form $\alpha$, by Theorem 5.1 the cohomology class it represents splits into a sum of elements of the form:

$$\begin{aligned}{}[\eta _{i_1}]\cdots [\eta _{i_q}][{\tilde{\alpha }}]\in [\eta _{i_1}]\cdots [\eta _{i_q}]H^{p}(M/{\mathcal {F}})\subset H^{p+q}_{dR}(M). \end{aligned}$$

hence we can write this harmonic form as the sum of the harmonic forms corresponding to such classes. The proof is now finished by noting that the forms $\eta _{i_1}\cdots \eta _{i_q}{\tilde{\alpha }}$ where ${\tilde{\alpha }}$ is basic harmonic are the representatives of such classes (as this implies that they are basic harmonic by the previous paragraph). $\square$

Theorem 6.3

Let $M^{2n+s}$ be a compact ${\mathcal {S}}$-manifold. A form $\alpha$ on M is harmonic if and only if it is a linear combination of elements of the form $(\eta _1-\eta _{i_1})\cdots (\eta _1-\eta _{i_q}){\tilde{\alpha }}$ such that ${\tilde{\alpha }}$ is basic primitive harmonic and their duals (via the star operator).

Proof

We start by noting that any element $\alpha$ of the given form is in fact harmonic (which immediately implies that the linear combination of such elements is also harmonic), since through straightforward computation (with the use of Proposition 2.28) we can get $d\alpha =\delta \alpha =0$.

On the other hand given any harmonic form $\alpha$, by Theorem 4.2 the cohomology class it represents splits into a sum of elements of one of the following forms:

$$\begin{aligned}{} & {} [\eta _1-\eta _{i_1}]\cdots [\eta _1-\eta _{i_q}][{\tilde{\alpha }}]\in [\eta _1-\eta _{i_1}]\cdots [\eta _1-\eta _{i_q}]{\mathcal {P}}H^{p}(M/{\mathcal {F}})\subset H^{p+q}_{dR}(M),\\{} & {} [\eta _1\eta _{i_1}\cdots \eta _{i_{q-1}}{\tilde{\alpha }}]\in \eta _1\eta _{i_1}\cdots \eta _{i_{q-1}}Ker^p(L)\subset H^{p+q}_{dR}(M). \end{aligned}$$

hence it suffices to find the harmonic representatives of such classes.

For the classes in $[\eta _1-\eta _{i_1}]\cdots [\eta _1-\eta _{i_q}]{\mathcal {P}}H^{p}(M/{\mathcal {F}})$ let us first note that by the Lefschetz decomposition of harmonic forms the basic harmonic representative of a class $[{\tilde{\alpha }}]\in {\mathcal {P}}H^p(M/{\mathcal {F}})$ is indeed primitive. Now it suffices to note that the class $[\eta _1-\eta _{i_1}]\cdots [\eta _1-\eta _{i_q}][{\tilde{\alpha }}]$ is indeed represented by $(\eta _1-\eta _{i_1})\cdots (\eta _1-\eta _{i_q}){\tilde{\alpha }}$ such that ${\tilde{\alpha }}$ is the basic harmonic representative of the basic class $[{\tilde{\alpha }}]$.

For the classes in $\eta _1\bigwedge \text {}^{\bullet }_{Ker(L)}<\eta _2,\ldots ,\eta _s>$ we shall prove that each such class is represented by a linear combination of duals of harmonic forms representing a class in $[\eta _1-\eta _{i_1}]\cdots [\eta _1-\eta _{i_q}]{\mathcal {P}}H^{p}(M/{\mathcal {F}})$. Let us start by taking the harmonic representative $(\eta _1-\eta _{i_1})\cdots (\eta _1-\eta _{i_q}){\tilde{\alpha }}$ of a class:

$$\begin{aligned}{}[\eta _1-\eta _{i_1}]\cdots [\eta _1-\eta _{i_q}][{\tilde{\alpha }}]\in [\eta _1-\eta _{i_1}]\cdots [\eta _1-\eta _{i_q}]{\mathcal {P}}H^{p}(M/{\mathcal {F}})\subset H^{p+q}_{dR}(M). \end{aligned}$$

We can see by Proposition 2.28 (and the fact that the basic star operator takes basic primitive harmonic forms to harmonic forms in Ker(L)) that its dual is an element of ${\overline{\bigwedge }}\text {}_{Ker(L)}^{\bullet } <\eta _{1},\ldots ,\eta _{s}>$. Hence, by Lemma 4.1 (and the proof of Theorem 4.2) it represents a class in:

$$\begin{aligned}{} & {} {\overline{\bigwedge }}\text {}_{Ker(L)}^{\bullet }<\eta _{1},\ldots ,\eta _{s}>/{\overline{\bigwedge }}\text {}_{Ker(L)}^{\bullet }<\eta _1-\eta _{2},\ldots ,\eta _1-\eta _{s}>\\{} & {} \quad \cong (\eta _1\bigwedge \text {}^{\bullet }_{Ker(L)}<\eta _2,\ldots ,\eta _s>)\subset H^{p+q}_{dR}(M). \end{aligned}$$

This shows that $*$ induces a morphism:

$$\begin{aligned}{} & {} {}[*]: \bigwedge \text {}^{\bullet }_{{\mathcal {P}}H^{\bullet }(M/{\mathcal {F}})}<[\eta _1-\eta _2],\ldots ,[\eta _1-\eta _s]>\rightarrow \eta _1\bigwedge \text {}^{\bullet }_{Ker(L)}<\eta _2,\ldots ,\eta _s>, \end{aligned}$$

by acting on the harmonic representatives. Moreover, due to well known Hodge theoretic results the star operator induces a bijection on harmonic forms and hence the morphism $[*]$ is a monomorphism (since it is a restriction of the star operator composed with the morphism induced by the inclusion of harmonic forms). From this we deduce that this morphism is in fact an isomorphism, since it is a monomorphism between vector spaces of the same (finite) dimension. This together with the fact that the star operator preserves harmonic forms shows that the harmonic representatives of the classes from $\eta _1\bigwedge \text {}^{\bullet }_{Ker(L)}<\eta _2,\ldots ,\eta _s>$ are precisely the duals of the harmonic forms given as linear combinations of elements of the form $(\eta _1-\eta _{i_1})\cdots (\eta _1-\eta _{i_q}){\tilde{\alpha }}$ such that ${\tilde{\alpha }}$ is basic primitive harmonic. $\square$

Remark 6.4

Again, similarly as in the previous section we believe that the methods used above can be useful in computing harmonic forms of general ${\mathcal {K}}$-manifolds on a case by case basis.

7 Application: stability of basic Betti and Hodge numbers

Here we wish to study the behaviour of the basic cohomology of an almost ${\mathcal {K}}$-manifold under deformations. Let us start by making the notion of a deformation of an almost ${\mathcal {K}}$-manifold precise.

Definition 7.1

Let $M^{2n+s}$ be a compact almost ${\mathcal {K}}$-manifold. A deformation $\{M_t\}_{t\in [0,1]}$ of M consists of the following data:

• (1) A (0, 2)-tensor g on $M\times [0,1]$ such that its restriction $g_t$ to each $M_t:=M\times \{t\}$ is a Riemannian metric and $g(\frac{\partial }{\partial t},\bullet )=0$,

• (2) A (1, 1)-tensor f on $M\times [0,1]$ which induces on each $M_t$ an f-structure $f_t$ such that $f_t(\frac{\partial }{\partial t})=0$,

• (3) Pointwise linearly independent vector fields $\{\xi _1,\ldots ,\xi _s\}$ on $M\times [0,1]$ which are tangent to M (again we denote their restriction to $M_t$ by $\xi _{kt}$),

such that each $M_t$ with the data induced on it is an almost ${\mathcal {K}}$-manifold and the structure induced on $M_0$ is precisely the initial structure on $M^{2n+s}$. We say that $M_t$ is a deformation of (almost) ${\mathcal {S}}$-structures or (almost) ${\mathcal {C}}$-structures if each $M_t$ is a (almost) ${\mathcal {S}}$-manifold or (almost) ${\mathcal {C}}$-manifold respectively.

Remark 7.2

• (1) We note that it is sufficient to specify the data given in the above definition to define an almost ${\mathcal {K}}$-structure on each $M_t$ as other data (such as the 2-form F) can be computed from it.

• (2) It is also important to note that in particular such a deformation is also a deformation of the transversely Kähler foliation.

We start by noting the following simple consequence of our main results:

Theorem 7.3

Let $\{M^{2n+s}_t\}_{t\in [0,1]}$ be a deformation of compact almost ${\mathcal {K}}$-manifolds and let $b^k_t:[0,1]\rightarrow {\mathbb {N}}$ be the function assigning to each t the k-th basic Betti number of $M_t$. Then:

• (1) If $M_t$ is a deformation of almost ${\mathcal {S}}$-manifolds such that each $M_t$ satisfies the basic hard Lefschetz property then the function $b^k_t$ is constant. In particular this is true for deformations of ${\mathcal {S}}$-manifolds,

• (2) If $M_t$ is a deformation of almost ${\mathcal {C}}$-manifolds then the function $b^k_t$ is constant.

Proof

Immediate since for all $t_1,t_2\in [0,1]$ the almost ${\mathcal {K}}$-manifolds $M_{t_1}$ and $M_{t_2}$ satisfy the assumption of Corollary 4.4 in the first case and Corollary 5.2 in the second case. $\square$

We now study the behaviour of basic Dolbeault cohomology using an approach similar to that of [12, 16]. We start by recalling a result from [16] which reduces the problem to proving that the spaces of complex-valued basic harmonic k-forms ${\mathcal {H}}^k_t$ of $({\mathcal {M}}_t,{\mathcal {F}}_t)$ form a bundle over [0, 1].

Theorem 7.4

Let $\{(M_t,{\mathcal {F}}_t)\}_{t\in [0,1]}$ be a smooth family of homologically orientable transversely Kähler foliations on compact manifolds such that ${\mathcal {H}}^k_t$ forms a smooth family of constant dimension for any $k\in {\mathbb {N}}$. For a fixed pair of integers (p, q) the function associating to each point $s\in [0,1]$ the basic Hodge number $h^{p,q}_t$ of $(M_t,{\mathcal {F}}_t)$ is constant.

Using this result the study can now be concluded analogously as in [16]. While the differences in the argument are scarce, we present it in full for the readers convenience following closely the exposition in [16].

The first step is to consider transverse k-forms. We denote the space of such forms by $\Omega ^{T,k}$. On such forms it is natural to consider the operator $d_T:=\pi (d)$ where $\pi$ is the projection onto transverse forms given by the Riemannian metric. Its adjoint $\delta _T$ is given by the formula:

$$\begin{aligned} \delta _T:=(-1)^k*_b^{-1} d_T *_{b}, \end{aligned}$$

which due to homological orientability coincides on basic forms with the basic coderivative $\delta _b$. This allows us to define the transverse Laplace operator in a fashion similar to [10, 12]:

$$\begin{aligned} \Delta ^T:=\sum \limits _{k=1}^s{\mathcal {L}}_{\xi _k}{\mathcal {L}}_{\xi _k}-\delta _Td_T-d_T\delta _T, \end{aligned}$$

and similarly as in [12, 16] we can prove the following lemma:

Lemma 7.5

The operator $\Delta ^T:\Omega ^{k,T}\rightarrow \Omega ^{k,T}$ is strongly elliptic and self-adjoint.

Proof

Around any point $x_0$ take a local coordinate chart $(z_1,\ldots ,z_s,x_1,y_1,\ldots ,x_n,y_n)$ where $\xi _k=\frac{\partial }{\partial z_k}$ and $(x_1,y_1,\ldots ,x_n,y_n)$ are transverse holomorphic coordinates such that $(\frac{\partial }{\partial x_1},\frac{\partial }{\partial y_1},\ldots \frac{\partial }{\partial x_n},\frac{\partial }{\partial y_n})$ are orthonormal over $x_0$ and $\eta _{k}=dz_{k}+\beta _k$ for some basic forms $\beta _k$ vanishing over $x_0$. In such coordinates the principal symbol $\sigma (\delta _{T}d_T+d_T\delta _T)$ coincide with that of the Laplacian $\Delta _b$ on the planes $z_1=\cdots =z_s=0$ (to see this note that in these coordinates $\pi (dz_k)=-\beta _k$ and so after writing the operator in local coordinates we see that aside from the parts present in $\Delta _b$ the additional components are either of degree less then 2 or are a multiple of some $\beta _k$ (which vanish over $x_0$) and hence in either case do not contribute to the symbol over $x_0$). For $\alpha :=\sum \limits _{i=1}^s\gamma _idz_i+\sum \limits _{i=1}^n \alpha _{2i-1}dx_i+\alpha _{2i}dy_i\in T^*_{x_0}M$ let $\sigma _{\alpha }(\Delta ^{T})$ be the symbol of $\Delta ^T$ at $\alpha$. The symbol $\sigma _{\alpha }(\frac{\partial ^2}{\partial ^2 z_k})=\gamma _k^2Id_{(\Omega ^{k,T})_{x_0}}$, while the symbol of $\Delta _b$ is given by $\sigma (\Delta _b)=-(\sum \limits _{i=1}^{2n}\alpha _i^2)Id_{(\Omega ^{k,T})_{x_0}}$ (see [18] Lemma 5.18). This shows that the symbol $\sigma _{\alpha }(\Delta ^T)=||\alpha ||^2Id_{(\Omega ^{k,T})_{x_0}}$ and so the operator is in fact strongly elliptic.

Since $\delta _{T}d_T+d_T\delta _T$ is self-adjoint it suffices to prove that each ${\mathcal {L}}_{\xi _k}$ is skew-symmetric. For $\alpha _1,\alpha _2\in \Omega ^{k,T}$ we have:

$$\begin{aligned} {\mathcal {L}}_{\xi _k}(\eta _1\wedge \cdots \wedge \eta _s\wedge \alpha _1\wedge *_b\overline{\alpha _2})= & {} \eta _1\wedge \cdots \wedge \eta _s\wedge {\mathcal {L}}_{\xi _k}(\alpha _1)\wedge *_b\overline{\alpha _2} +\eta _1\wedge \cdots \wedge \eta _s\\&\wedge&\alpha _1\wedge *_b{\mathcal {L}}_{\xi _k}\overline{\alpha _2}, \end{aligned}$$

since ${\mathcal {L}}_{\xi _k}\eta _l=0$ and ${\mathcal {L}}_{\xi _k}*_b=*_b{\mathcal {L}}_{\xi _k}$. Hence, we only need to prove that the left hand side integrates to zero over M. But we can write it as:

$$\begin{aligned} di_{\xi _k}(\eta _1\wedge \cdots \wedge \eta _s\wedge \alpha _1\wedge *_b{{\overline{\alpha }}}_2)=d(\eta _1\wedge \cdots \wedge \eta _{k-1}\wedge \eta _{k+1}\wedge \cdots \wedge \eta _s\wedge \alpha _1\wedge *_b{{\overline{\alpha }}}_2). \end{aligned}$$

Now it suffices to note that the right hand-side is exact and hence integrates to zero. $\square$

With this we are ready to prove the following result:

Proposition 7.6

Let $\{M_t\}_{t\in [0,1]}$ be a smooth family of ${\mathcal {C}}$-manifolds or ${\mathcal {S}}$-manifolds over an interval. Then the spaces ${\mathcal {H}}^k_t$ of complex-valued basic harmonic k-forms on $M_t$ constitute a bundle over [0, 1].

Proof

We start by using the results of [13] in a fashion similar to [12] and [16] in order to contain our problem in some smooth vector bundle (with fibers of finite dimension). Using the Spectral Theorem for smooth families of strongly elliptic self-ajoint operators (see Theorem 1 of [13]) for the family $\Delta ^{k,T}_t$ we get a complete system of eigensections $\{e_{th}\}_{h\in {{\mathbb {N}}}, t\in [0,1]}$ together with the corresponding eigenvalues $\lambda _h(t)$ which form an ascending sequence in $[0,\infty )$ with a single accumulation point at infinity. Fix a point $t_0\in [0,1]$ and let $k_0$ be the largest number such that for $h\in \{1,\ldots ,k_0\}$ we have $\lambda _h(t_0)=0$. Consider the family of vector spaces ${\mathcal {E}}_t=span\{e_{th}\text { }|\text { }h\in \{1,\ldots ,k_0\}\}$. Since the only accumulation point of the sequence $\lambda _{h}(t_0)$ is infinity we can find a small disc around 0 in ${\mathbb {C}}$ such that the only eigenvalue of $\Delta ^{k,T}_{t_0}$ contained in this disc is zero. Using Theorem 2 of [13] we establish that for each h the eigenvalues $\lambda _h(t)$ form a continuous function and hence in a small neighbourhood U of $t_0$ all $t\in U$ are contained in this disc as well. This allows us to conclude by using Theorem 3 of [13] that $P_{{\mathcal {E}}_t}({\tilde{e}}_{th})$ for $h\in \{1,\ldots ,k_0\}$ form smooth sections of $\Omega ^{k,T}$ over a small neighbourhood $U'\subset U$ of $t_0$ which span ${\mathcal {E}}_t$ (where $P_{{\mathcal {E}}_t}$ is the projection onto ${\mathcal {E}}_t$ and ${\tilde{e}}_{th}$ are the extensions of $e_{t_0h}$ with the use of some partition of unity over [0, 1]). Shrinking the neighbourhood is necessary to retain linear independence of ${\tilde{e}}_{th}$. Hence, we have shown that ${\mathcal {E}}_t$ form a bundle over $U'$.

Now we consider the operator ${\mathcal {L}}_t=({\mathcal {L}}_{\xi _1t},\ldots ,{\mathcal {L}}_{\xi _st}):{\mathcal {E}}_t\rightarrow \bigoplus \limits _{i=1}^s\Omega ^{k,T}_t$. Note that $Ker{\mathcal {L}}_{t_0}={\mathcal {H}}^k_{t_0}$. Via a standard rank argument there is a small neighbourhood $U''\subset U'$ of $t_0$ such that $dim(Ker{\mathcal {L}}_{t_0})\ge dim(Ker{\mathcal {L}}_{t})$. However, $Ker{\mathcal {L}}_{t}\supset {\mathcal {H}}^k_t$ and since $dim({\mathcal {H}}^k_t)=dim({\mathcal {H}}^k_{t_0})$ (by Theorem 7.3) we have the following:

$$\begin{aligned} dim(Ker{\mathcal {L}}_{t_0})\ge dim(Ker{\mathcal {L}}_{t})\ge dim({\mathcal {H}}^k_t)=dim({\mathcal {H}}^k_{t_0})=dim(Ker{\mathcal {L}}_{t_0}). \end{aligned}$$

hence all of the dimensions above are equal and $Ker{\mathcal {L}}_{t}= {\mathcal {H}}^k_t$. But this implies that ${\mathcal {H}}^k_s$ can be described as a kernel of a morphism of bundles and since its dimension is constant we conclude that it is a bundle (over $U''$). It immediately follows that ${\mathcal {H}}^k_t$ forms a bundle over [0, 1] since it is a family of subspaces of a bundle with local trivializations around any point. $\square$

Combining Theorem 7.4 and Proposition 7.6 we get the main result of this section:

Theorem 7.7

Let $\{M^{2n+s}_t\}_{t\in [0,1]}$ be a deformation of compact ${\mathcal {C}}$-manifolds or ${\mathcal {S}}$-manifolds. Then the function $h^{p,q}_t:[0,1]\rightarrow {\mathbb {N}}$, assigning to each t the (p, q)-th basic Hodge number of $M_t$, is constant.