Vertical–horizontal decomposition of Laplacians and cohomologies of manifolds with trivial tangent bundles

Abstract

In this paper, we obtain a vertical–horizontal decomposition formula of Laplacians on manifolds with a special foliation structure. Two Nomizu-type theorems for cohomologies of nilmanifolds follow as applications.

1 Introduction

Let X be an n-dimensional compact smooth manifold. Assume that the tangent bundle $T_X$ of X is trivial (thus the cotangent bundle $T^*_X$ is also trivial). Let

$$\begin{aligned} \Phi :=\{\sigma ^1, \ldots , \sigma ^n\} \end{aligned}$$

be a global smooth frame of $T^*_X$. Inspired by [9, 20], we introduce the following

Definition 1.1

We call $\Phi$ a nilpotent frame if

$$\begin{aligned} d\sigma ^j =\sum _{k,l>j} A_{kl}^j \, \sigma ^k \wedge \sigma ^l, \qquad \forall \ 1\le j\le n, \end{aligned}$$

(1.1)

where $A_{kl}^j$ are “real constants.”

In the complex case, assume that the holomorphic tangent bundle $\wedge ^{1,0} T_X$ of a compact complex manifold X is smoothly trivial (may not be trivial as a holomorphic vector bundle). Now assume that the complex dimension of X is n. Let

$$\begin{aligned} \Psi :=\{\xi ^1, \ldots , \xi ^n\} \end{aligned}$$

be a global smooth frame of $\wedge ^{1,0}T^*_X$. We shall use the following

Definition 1.2

We call $\Psi$ a complex nilpotent frame if

$$\begin{aligned} d\xi ^j =\sum _{k,l>j} B_{kl}^j \, \xi ^k \wedge \xi ^l+\sum _{k,l>j} B_{k\bar{l}}^j \, \xi ^k \wedge \overline{\xi ^l}, \qquad \forall \ 1\le j\le n, \end{aligned}$$

(1.2)

where $B_{kl}^j$ and $B_{k\bar{l}}^j$ are “complex constants.”

We have the following generalization of the main results in [9, 13, 20].

Main Theorem

Let X be a compact smooth (resp. complex) manifold. Assume that $T^*_X$ (resp. $\wedge ^{1,0} T^*_X$ ) possesses a nilpotent (resp. complex nilpotent) frame $\Phi$ (resp. $\Psi$). Then, every de Rham (resp. Dolbeault) cohomology class of X can be represented by $\mathbb {R}$ (resp. $\mathbb {C}$) linear combination of finite wedge products of forms in $\Phi$ (resp. $\Psi \cup \overline{\Psi }$).

Remark 1

Denote by $A^{\star }$ (resp. $A^{\star , \star }$) the finite dimension $\mathbb {R}$ (resp. $\mathbb {C}$) linear space spanned by wedge products of $\Phi$ (resp. $\Psi \cup \overline{\Psi }$). Then we know that the d-cohomology is well defined on $A^{\star }$, the $\overline{\partial }$-cohomology is well defined on $A^{\star , \star }$ and they are also called the Lie algebra cohomologies. Let us denote them by $H^{\star }_{d, \Phi }$ and $H^{\star , \star }_{\overline{\partial }, \Psi }$, respectively. Then, our main theorem is equivalent to say that

$$\begin{aligned} H_d^{\star } \simeq H^{\star }_{d, \Phi }, \ \ \ H_{\overline{\partial }}^{\star , \star } \simeq H^{\star , \star }_{\overline{\partial }, \Psi }, \end{aligned}$$

where $H_d^{\star }$ (resp. $H_{\overline{\partial }}^{\star , \star }$) denotes the usual de Rham (resp. Dolbeault) cohomology group. See Sect. 8 for a more explicit description of $H^{\star }_{d, \Phi }$ and $H^{\star , \star }_{\overline{\partial }, \Psi }$ in certain cases and Sect. 9 for related results.

Remark 2

The main ingredient in the proof of our main theorem is the following vertical–horizontal decomposition of Laplacians (see Theorem 4.1)

$$\begin{aligned} \Box _d=\Box _{d^v}+\Box _{d^h+R_d}, \end{aligned}$$

associated to the following decomposition

$$\begin{aligned} d=d^h+d^v+R_d, \end{aligned}$$

of d on a smooth manifold with a special foliation structure, where $d^h$ only increases the horizontal degree, $d^v$ only increases the vertical degree and the remaining term $R_d$ is a tensor (see [1,2,3, 18, 19, 21, 26] for the background and related results).

Remark 3

The proof of our main theorem in Sect. 7 also gives the following result: Let X be a compact smooth manifold. Assume that $\mathbb {C}\otimes T^*_X$ possesses a nilpotent frame $\Phi$. Then every complex de Rham cohomology class of X can be represented by $\mathbb {C}$ linear combination of finite wedge products of forms in $\Phi$.

Our main theorem suggests to study the following problem:

$(\star )$: LetGbe a Lie group, let$\Gamma$be a discrete subgroup ofG. Assume that with respect to the left action of$\Gamma$, $X:=\Gamma \backslash G$is a compact manifold. When does$T^*_X$possess a nilpotent frame ?

If G is nilpotent then of course $T^*_X$ possesses a nilpotent frame. But it is also interesting to study the general case, e.g., $\mathrm{SL}(2,\mathbb {Z})\backslash \mathrm{SL}(2, \mathbb {R})$ (non-compact!). In Sect. 9, we shall give an example where $T^*_X$ possesses a nilpotent frame but G is not nilpotent. For related results, see [6, 7].

2 Motivations

2.1 First motivation: Künneth formula

Our first motivation comes from the following well-known Künneth formula:

Theorem 2.1

(Künneth formula) Let $(X, g_X)$ and $(Y, g_Y)$ be two compact Riemannian manifolds. Let $(E, h_E)$, $(F, h_F)$ be Hermitian complex vector bundles over X and Y, respectively.

• If E and F are flat, then we have the following formula for de Rham cohomologies:

  $$\begin{aligned} H^k_d(X\times Y, E\otimes F)=\oplus _{p+q=k} H^p_d(X, E)\otimes H^q_d(Y, F); \end{aligned}$$

• If X, Y are complex manifolds and E, F are holomorphic vector bundles, then

  $$\begin{aligned} H^{p,q}_{\overline{\partial }}(X\times Y, E\otimes F)=\oplus _{j+k=p, l+m=q} H^{j,l}_{\overline{\partial }}(X, E)\otimes H_{\overline{\partial }}^{k,m}(Y, F). \end{aligned}$$

One way to prove the above formulas is to use the Leray spectral sequence for fibrations, see [9]. Our motivation comes from the proof of using the following decomposition formulas of Laplacians:

$$\begin{aligned} \Box _d^{X\times Y}=\Box _d^X+\Box _d^Y, \end{aligned}$$

(2.1)

and

$$\begin{aligned} \Box _{\overline{\partial }}^{X\times Y}=\Box _{\overline{\partial }}^X+\Box _{\overline{\partial }}^Y. \end{aligned}$$

(2.2)

More precisely, we will study the following problem:

Problem: How to generalize (2.1) and (2.2) to non-product fibrations?

Remark

One way to study the above problem is to develop the $L^2$-theory of the Leray spectral sequence for fibrations (see [5, 11] for related results). We know that for the spectral sequence of the double complex $(\partial , \overline{\partial })$, $d=\partial +\overline{\partial }$, the associated $L^2$-theory is based on the classical Bochner–Kodaira–Nakano formula.

2.2 Second motivation: Nomizu-type theorems

Our second motivation is based on the following celebrated Nomizu’s theorem [20] proved in 1954:

Nomizu’s theorem (weak version)LetGbe a simply connected nilpotent Lie group with a discrete subgroup$\Gamma$. Assume that$X:=\Gamma \backslash G$is compact and the ascending central series of the Lie algebra ofG(see Sect. 6.2for the definition) defines a torus fibration resolution ofX. Then the de Rham cohomology ofXcan be represented byG-invariant forms.

In 1976, Sakane [24] proved that the Nomizu theorem is also true for compact complex parallelisable solvmanifolds. The following theorem of Cordero–Fernández–Gray–Ugarte [9] is a generalization of Sakane’s theorem:

Cordero–Fernández–Gray–Ugarte’s theorem (weak version)LetGbe a simply connected nilpotent Lie group with a discrete subgroup$\Gamma$. Assume that$X:=\Gamma \backslash G$is a compact manifold with a left invariant integrable almost complex structureJ. Assume that theJ-compatible ascending series of the Lie algebra ofG(see Sect. 6.3 for the definition) defines a holomorphic torus fibration resolution ofX. Then, the Dolbeault cohomology ofXcan be represented byG-invariant forms.

Remark

The assumption that the J-compatible ascending series of the Lie algebra of G defines a holomorphic torus fibration resolution is contained in the proof of the main theorem in [9].

In real case the ascending central series will always give a torus fibration resolution (see page 208 in [10]). Thus the following result is still true:

Nomizu’s theorem (original version)LetGbe a simply connected nilpotent Lie group with a discrete subgroup$\Gamma$. Assume that$X:=\Gamma \backslash G$is compact. Then, the de Rham cohomology ofXcan be represented byG-invariant forms.

In complex case, the J-compatible ascending series may not give a torus fibration resolution (see Example 3.6 in [23] or [13]). But our main theorem implies the following result.

Cordero–Fernández–Gray–Ugarte’s theorem (strong version)LetGbe a simply connected nilpotent Lie group with a discrete subgroup$\Gamma$. Assume that$X:=\Gamma \backslash G$is a compact manifold with a nilpotent complex structure (see [9], page 2, for the definition). Then, the Dolbeault cohomology ofXcan be represented byG-invariant forms. In particular, it is independent of$\Gamma$.

The above result applies in a number of important cases.

Corollary 2.2

Let G be a simply connected nilpotent Lie group with a discrete cocompact subgroup $\Gamma$ and left-invariant complex structure J. If G is 2-step nilpotent, then the Dolbeault cohomology of $X = (\Gamma \backslash G,J)$ can be computed by left-invariant forms.

Proof

We only have to observe that if G is 2-step nilpotent, then every left-invariant complex structure on G is nilpotent in the above sense by [22, Prop. 3.3] so the strong version of Cordero–Fernández–Gray–Ugarte’s theorem applies. $\square$

A different way to generalize the weak version of Cordero–Fernández–Gray–Ugarte’s theorem was considered in [13] and like in loc. cit. we are able to settle all cases of low dimension.

Corollary 2.3

Let X be a nilmanifold of real dimension at most 6 with left-invariant complex structure. Then, the Dolbeault cohomology of X is computed by left-invariant forms.

Proof

In dimension 2 and 4, there are only tori and the Kodaira–Thurston manifold to consider, for which the result is well known.

In real dimension 6, there are only finitely many nilpotent Lie algebras and the ones admitting complex structures are classified by Salamon [25]. In [22, Proof of Thm. B] the statement was shown to hold for all complex structures on all such nilmanifolds except possibly for those with Lie algebra $\mathfrak h_7$, in the notation of Salamon (see also [8] for the original definition). Since $\mathfrak h_7$ is 2-step nilpotent, indeed the free 2-step nilpotent Lie algebra on 3 generators, the previous corollary applies to this remaining case. $\square$

3 Foliations of nilpotent type

3.1 Nilpotent foliation

Let us recall the definition of distribution first.

Definition 3.1

(Distribution) Let X be a smooth manifold. We call

$$\begin{aligned} \mathcal {V}:=\{\mathcal {V}_x\}_{x\in X}, \end{aligned}$$

a rank-rdistribution on X if for every $x\in X$, $\mathcal {V}_x$ is an r-dimensional real linear subspace of $T_x X$ (space of vectors at x) and there exist smooth vector fields $V_1, \ldots , V_r$ on an open neighborhood, say $U_x$, of x such that

$$\begin{aligned} \mathcal {V}_y=\mathrm{Span}_{\mathbb {R}} \{V_1(y), \ldots , V_r(y)\}, \end{aligned}$$

for every $y\in U_x$. We call $\{V_1, \ldots , V_r\}$ a local basis of $\mathcal {V}$.

Remark

If V is a smooth vector field on X such that $V(x)\in \mathcal {V}_x$ for every $x\in X$, then we say that V lies in $\mathcal {V}$ and write $V\in \mathcal {V}$. Denote by $C^\infty (T_X)$ the space of smooth vector fields on X. Then, one may look at a rank-r distribution as a subspace of $C^\infty (T_X)$ that is locally generated by r linearly independent smooth vector fields.

Definition 3.2

(Integrable distribution) A distribution $\mathcal {V}$ is said to be integrable if $[V, W]\in \mathcal {V}$, for every $V, W\in \mathcal {V}$ (see the remark above). We call an integrable distribution a foliation on X.

Remark

It is enough to check integrability for local basis of $\mathcal {V}$. The classical Frobenius theorem tells us that a rank-r distribution $\mathcal {V}$ is integrable if and only if for every $x\in X$ there exists a smooth local coordinate system, say $\{x^1, \ldots , x^n\}$, near x such that $\mathcal {V}$ is generated by $\{\partial /\partial x^1, \ldots , \partial /\partial x^r\}$ near x (i.e., $\mathcal {V}$ is tangent to the fibers of the map $(x^1, \ldots , x^n)\mapsto (x^{r+1}, \ldots , x^n)$). Thus, a rank-r integrable distribution is equivalent to a foliation of r-dimensional local smooth manifolds.

We shall use the following lemma to define the notion of nilpotent foliation.

Lemma 3.3

Let $\mathcal {V}$ be a distribution on X. Let $g_X$ be a smooth Riemannian metric on X. Then

$$\begin{aligned} \mathcal {V}^\bot :=\{\mathcal {V}_x^\bot \}_{x\in X}, \end{aligned}$$

is also a distribution on X, where each $\mathcal {V}_x^\bot$ denote the orthogonal complement of $\mathcal {V}_x$ in $T_xX$ with respect to $g_X$.

Proof

Let $\{V_1, \ldots , V_r\}$ be a local basis of $\mathcal {V}$. Then we can extend it to a local frame, say $\{V_1, \ldots , V_n\}$, of $T_X$. Denote by $V_j^\bot$, $j>r$, the orthogonal projection of $V_j$ to $\mathcal {V}^\bot$. Then, we know that $\{V_j^\bot \}_{j>r}$ generates $\mathcal {V}^\bot$ locally. $\square$

Definition 3.4

(Nilpotent foliation) Let $\mathcal {V}$ be a distribution on a Riemannian manifold $(X, g_X)$. We call $(\mathcal {V}, g_X)$ a nilpotent foliation structure on X if locally there exists an orthonormal frame $\{V_1, \ldots , V_n\}$ of $(T_X, g_X)$ such that

• (1) $\{V_j\}_{j\le r}$ is a local basis of $\mathcal {V}$ and $\{V_j\}_{j> r}$ is a local basis of $\mathcal {V}^\bot$;

• (2) $[V_j, V_k]=0$ for every $1\le j\le r$, $1\le k\le n$.

Remark

Notice that condition (2) in the above definition implies that a nilpotent foliation is always integrable.

We shall also study nilpotent foliations on complex manifold.

Definition 3.5

(Complex nilpotent foliation) Let $\mathcal {V}$ be a distribution on a complex manifold (X, J). Let $g_X$ be a J-Hermitian metric on X. We call $(\mathcal {V}, J, g_X)$ a complex nilpotent foliation structure on X if locally there exists an orthonormal frame $\{V_1, \ldots , V_n\}$ of $(T^{1,0}_X, g_X)$ such that

• (1) $\{V_j, \bar{V}_j\}_{j\le r}$ is a local basis of $\mathcal {V}$ and $\{V_j, \bar{V}_j\}_{j> r}$ is a local basis of $\mathcal {V}^\bot$;

• (2) $[V_j, V_k]=[V_j, \bar{V}_k]=0$ for every $1\le j\le r$, $1\le k\le n$.

Remark

Since $g_X$ is J-Hermitian, a complex nilpotent foliation also satisfies $J(\mathcal {V}^\bot )=\mathcal {V}^\bot$.

3.2 Vertical–horizontal decomposition of d

Definition 3.6

(Vertical–horizontal vector field) Let $\mathcal {V}$ be a distribution on a Riemannian manifold $(X, g_X)$. We call $V\in \mathcal {V}$ a vertical vector field and $W\in \mathcal {V}^\bot$ a horizontal vector field.

We also need the dual of the notion of vertical–horizontal vector field (motivated by [4], see also formula (1.3) in [1]).

Definition 3.7

(Vertical–horizontal one-form) A differential one-form u on X is said to be horizontal (resp. vertical) if $V\, \rfloor \, u=0$ for every vertical (resp. horizontal) vector field V on X.

Definition 3.8

(Vertical–horizontal degree) Denote by $T^*_h$ and $T^*_v$ the subbundles of $T^*X$ generated by horizontal one-forms and vertical one-forms, respectively. Then, we have

$$\begin{aligned} \wedge ^p T^*X=\oplus _{k+l=p} (\wedge ^k T^*_h) \wedge (\wedge ^l T^*_v). \end{aligned}$$

We call a section of $(\wedge ^k T^*_h) \wedge (\wedge ^l T^*_v)$ a degree (k|l)-form and say that it has horizontal degreek and vertical degreel.

The following lemma suggests to study vertical–horizontal decomposition of the exterior derivative.

Lemma 3.9

Let $\mathcal {V}$ be a distribution on a Riemannian manifold $(X, g_X)$. Let u be a smooth degree (k|l)-form on X. Assume that $\mathcal {V}$ is integrable. Then, we can write

$$\begin{aligned} du=d^v u+d^h u +R_d u, \end{aligned}$$

where $d^v u$ is degree $(k|l+1)$, $d^h u$ is degree $(k+1|l)$ and $R_d u$ is degree $(k+2| l-1)$.

Proof

Let us locally write

$$\begin{aligned} u=\sum u_h^j \wedge u_v^j, \end{aligned}$$

where $u_v^j$ are (0|l)-forms and $u_h^j$ are (k|0)-forms. Since $\mathcal {V}$ is integrable, we know that $d(u_h^j)$ has no degree $(k-1|2)$ components. Thus, du has no degree $(k-1|l+2)$ components. $\square$

Definition 3.10

(Atiyah tensor) Let $\mathcal {V}$ be an integrable distribution on a Riemannian manifold $(X, g_X)$. Then, we define $d^h$ as the (1|0)-part of d and $d^v$ as the (0|1)-part of d. We call the following degree $(2|-1)$ tensor

$$\begin{aligned} R_d:=d-d^h-d^v, \end{aligned}$$

the Atiyah tensor.

Remark 1

From the proof of the above Lemma, we know that the Atiyah tensor is zero if and only if $\mathcal {V}^\bot$ is integrable. In case $\mathcal {V}$ is associated to the Lie algebra $\mathfrak {g}$ of a G–bundle, then cohomology class of each Lie–algebra component of the Atiyah tensor is also called the Atiyah class.

Remark 2

$d^h, d^v$ are also well defined on the space of all smooth forms on X. The reason is we can always write a smooth form u as

$$\begin{aligned} u=\sum u^{(k|l)}, \end{aligned}$$

where each $u^{(k|l)}$ denotes the degree (k|l)-component of u. Then we can define

$$\begin{aligned} d^h u=\sum d^hu^{(k|l)}, \ \ d^v u=\sum d^v u^{(k|l)}. \end{aligned}$$

3.3 Vertical–horizontal decomposition of $\overline{\partial }$

Now let us consider the case that $\mathcal {V}$ is a distribution on a complex manifold (X, J) with a J-Hermitian Riemannian metric $g_X$ (we call $(X, J, g_X)$ a Hermitian complex manifold) such that $J(\mathcal {V})=\mathcal {V}$. Then we have

$$\begin{aligned} \wedge ^{p,q} T^*X=\oplus _{k+j=p, l+s=q} (\wedge ^{k,l} T^*_h) \wedge (\wedge ^{j,s} T^*_v). \end{aligned}$$

We call smooth section of $(\wedge ^{k,l} T^*_h) \wedge (\wedge ^{j,s} T^*_v)$ a degree (k, l|j, s)-form and say that it has horizontal degree (k, l) and vertical degree (j, s). Similar as the real case, we have

Lemma 3.11

Let $\mathcal {V}$ be an integrable distribution on a hermitian complex manifold $(X, J, g_X)$. Let u be a smooth degree (k, l|j, s)-form on X. Assume that $J(\mathcal {V})=\mathcal {V}$. Then we can write

$$\begin{aligned} \overline{\partial }u=\overline{\partial }^v u+\overline{\partial }^h u +R_{A_1} u +R_{A_2} u+ R_{KS} u, \end{aligned}$$

where $\overline{\partial }^v u$ is degree $(k,l|j,s+1)$, $\overline{\partial }^h u$ is degree $(k,l+1|j,s)$ and $R_{A_1} u$ is degree $(k+1,l+1|j-1,s)$, $R_{A_2} u$ is degree $(k,l+2|j,s-1)$ and $R_{KS} u$ is degree $(k+1, l| j-1,s+1)$.

Definition 3.12

(Complex Atiyah Tensor and Kodaira–Spencer Tensor) Let $\mathcal {V}$ be a J-invariant integrable distribution on a Hermitian complex manifold $(X, J, g_X)$. We define $\overline{\partial }^h$ as the (0, 1|0, 0)-part of $\overline{\partial }$ and $\overline{\partial }^v$ as the (0, 0|0, 1)-part of $\overline{\partial }$. We call

$$\begin{aligned} R_{A}:=R_{A_1}+R_{A_2}, \end{aligned}$$

the complex Atiyah tensorand $R_\mathrm{KS}$ the Kodaira–Spencer tensor.

Remark

In case $\mathcal {V}$ is given by the fiber-tangent distribution of a proper holomorphic submersion, then cohomology class of each component of $R_{KS}$ is just the well known Kodaira–Spencer class. In general, put

$$\begin{aligned} R_{\overline{\partial }}=R_A+R_{KS}, \end{aligned}$$

If $R_{KS}\ne 0$ then $R_d\ne R_{\overline{\partial }} +R_{\partial }$. In fact, we have

$$\begin{aligned} R_d=R_{A}+\overline{R_A}, \ \ d^v=\overline{\partial }^v+\partial ^v, \end{aligned}$$

and

$$\begin{aligned} d^h=R_\mathrm{KS}+\overline{R_\mathrm{KS}}+\partial ^h+\overline{\partial }^h. \end{aligned}$$

In case $\mathcal {V}$ is a complex nilpotent foliation, we can prove that

Lemma 3.13

Assume that $(\mathcal {V}, J, g_X)$ is a complex nilpotent foliation structure. Then $R_{KS}\equiv 0$.

Proof

It suffices to show that if u is an vertical (1, 0)-form then $\overline{\partial }u$ has no degree (1, 0|0, 1)-component. Since

$$\begin{aligned} du(W, V)=L_W(u(V))-L_V(u(W))-u([W, V]), \end{aligned}$$

(3.1)

it is enough to show that for every vertical (0, 1)-vector field V and horizontal (1, 0)-vector field W, the vertical (1, 0)-component of [V, W] is zero, which follows from 2) in Definition 3.5. $\square$

4 Vertical–horizontal decomposition of Laplacians

4.1 Fundamental theorem

The fundamental theorem in this paper is the following:

Theorem 4.1

(Real case): Let $(X, g_X)$ be an oriented Riemannian manifold with a nilpotent foliation structure (see Definition 3.4). Then on the space of smooth forms on X, we have

$$\begin{aligned} \Box _d=\Box _{d^v}+\Box _{d^h+R_d}. \end{aligned}$$

(4.1)

(Complex case): Let $(X, J, g_X)$ be a hermitian complex manifold with a complex nilpotent foliation structure (see Definition 3.5). Then, on the space of smooth forms on X, we have

$$\begin{aligned} \Box _{\overline{\partial }}=\Box _{{\overline{\partial }}^v}+\Box _{{\overline{\partial }}^h+R_{\overline{\partial }}}, \ \Box _{\overline{\partial }^v}=\Box _{\partial ^v}. \end{aligned}$$

(4.2)

Remark 1

In our proof, we shall use the following notation: if P is a differential operator on the space of smooth forms on X, then we shall write $P^*$ as the adjoint of P and write

$$\begin{aligned} \Box _P:=PP^*+P^*P. \end{aligned}$$

Recall that $P^*$ satisfies

$$\begin{aligned} (Pu, v)=(u, P^*v), \end{aligned}$$

if u is a smooth form and v is a smooth form with compact support. Thus, $P^*$ and $\Box _P$ are well defined on the space of smooth forms. If P maps a degree k form to a degree $(k+p)$ form, then we say that P has degree p. If P is a degree p operator and Q is a degree q operator, then we write

$$\begin{aligned} {[}P, Q]:=PQ-(-\,1)^{pq} QP. \end{aligned}$$

Since $d, d^v, d^h+R_d$ are degree one operators, we have

$$\begin{aligned} \Box _{d}=[d, d^*], \ \Box _{d^v}=[d^v, (d^v)^*], \end{aligned}$$

and

$$\begin{aligned} \Box _{d^h+R_d}=[d^h+R_d, (d^h+R_d)^*]. \end{aligned}$$

Thus (4.1) is equivalent to

$$\begin{aligned}{}[d^h, (d^v)^*]=0, \ \ [d^v, R_d^*]=0. \end{aligned}$$

(4.3)

Remark 2

Notice that if X is compact, then

$$\begin{aligned} ((\Box _P +\Box _Q) u, u)=||Pu||^2+||P^*u||^2+||Qu||^2+||Q^*u||^2, \end{aligned}$$

for every smooth form u on X. Thus $(\Box _P +\Box _Q) u=0$ is equivalent to

$$\begin{aligned} Pu=P^*u=Qu=Q^* u=0, \end{aligned}$$

which gives the following corollary:

Corollary 4.2

(Real case): Let $(X, g_X)$ be an oriented compact Riemannian manifold with a nilpotent foliation structure (see Definition 3.4). Then, a smooth form u lies in $\ker \Box _d$ if and only if

$$\begin{aligned} d^v u=(d^v)^*u=(d^h+R_d)u=(d^h+R_d)^*u=0 \ \ \text {on}\ X. \end{aligned}$$

(4.4)

(Complex case): Let $(X, J, g_X)$ be a hermitian compact complex manifold with a complex nilpotent foliation structure (see Definition 3.5). Then, a smooth form u lies in $\ker \Box _{\overline{\partial }}$ if and only if

$$\begin{aligned} {\overline{\partial }}^v u= ({\overline{\partial }}^v )^*u =({\overline{\partial }}^h+R_{\overline{\partial }}) u=({\overline{\partial }}^h+R_{\overline{\partial }})^* u= \partial ^v u= ( \partial ^v)^*u=0 \ \ \text {on}\ X. \end{aligned}$$

(4.5)

4.2 Proof of the real case

Let $(\mathcal {V}, g_X)$ be a nilpotent foliation structure on X (see Definition 3.4). Let $\{V_j\}$ be the local frame of $(T_X, g_X)$ in Definition 3.4. Let us write

$$\begin{aligned} X_j^h=V_{r+j}, \ \ X_k^v=V_k, \qquad 1\le k\le r, \ \ 1\le j\le n-r. \end{aligned}$$

We know each $X_k^v$ is vertical and each $X_j^h$ is horizontal. Denote by

$$\begin{aligned} \{\varphi ^j_h, \varphi ^k_v\}_{1\le k\le r, 1\le j\le n-r}, \end{aligned}$$

the dual frame of $\{X_j^h, X_k^v\}$. By 2) in Definition 3.4 and (3.1), we have

$$\begin{aligned} d\varphi ^j_h=\sum _{k,l=1}^{n-r} C_{kl}^j \, \varphi ^k_h \wedge \varphi ^l_h, \ \ 1\le j\le n-r, \end{aligned}$$

and

$$\begin{aligned} d\varphi ^k_v=\sum _{j,l=1}^{n-r} D_{jl}^k \, \varphi ^j_h \wedge \varphi ^l_h, \ \ 1\le k\le r, \end{aligned}$$

where $C_{kl}^j$ and $D_{jl}^k$ are smooth functions. Thus we have

Lemma 4.3

The components $R_d, d^v, d^h$ of d can be written as

$$\begin{aligned} R_d= & {} \sum _{k=1}^r \sum _{j,l=1}^{n-r} \left( D_{jl}^k \, \varphi ^j_h \wedge \varphi ^l_h\right) \wedge (X_k^v \,\rfloor \,),\\ d^v= & {} \sum _{p=1}^r \varphi _v^p \wedge (X_p^v),\\ d^h= & {} \sum _{j=1}^{n-r} \varphi _h^j \wedge (X_j^h) +\sum _{j,k,l=1}^{n-r} C_{kl}^j \, \left( \varphi ^k_h \wedge \varphi ^l_h\right) \wedge (X_j^h\, \rfloor \,). \end{aligned}$$

Now let us finish the proof of the first identity in (4.3).

Lemma 4.4

$[d^v, R_d^*]=0$.

Proof

Since $R_d$ is a tensor, we have

$$\begin{aligned} R_d^*=\sum _{k=1}^r \sum _{j,l=1}^{n-r} D_{jl}^k\, \varphi _v^k\wedge (X_l^h\, \rfloor \,)(X_j^h\, \rfloor \,). \end{aligned}$$

(4.6)

Thus $R_d^*$ commutes with $\varphi _v^p \wedge$. Moreover, $d^2\varphi _v^k=0$ gives that $d^v D_{jl}^k\equiv 0$. Thus $[d^v, R_d^*]=0$. $\square$

We need the following proposition to prove $[d^h, (d^v)^*]=0$.

Proposition 4.5

Denote by $*$ the Hodge star operator on our oriented manifold $(X, g_X)$. Assume that the orientation of X is given by $\Omega _h \wedge \Omega _v$, where

$$\begin{aligned} \Omega _h:=\varphi _h^1\wedge \cdots \wedge \varphi _h^{n-r}, \ \ \Omega _v:=\varphi _v^1\wedge \cdots \wedge \varphi _v^{r}. \end{aligned}$$

Denote by $*_h$ (resp. $*_v$) the Hodge star operator with respect to $\Omega _h$ (resp. $\Omega _v$) on the space of horizontal (resp. vertical) forms, respectively. Then

$$\begin{aligned} *(u_h \wedge u_v)=(-\,1)^{(n-r-p)q} *_hu_h\wedge *_v u_v, \end{aligned}$$

(4.7)

where $u_h$ is a degree p horizontal form, $u_v$ is a degree q vertical form.

Proof

Notice that

$$\begin{aligned} (u_h \wedge u_v) \wedge (-\,1)^{(n-r-p)q}(*_h u_h\wedge *_v u_v){=} (u_h \wedge *_h u_h) \wedge (u_v\wedge *_v u_v){=}(u_h \wedge u_v)\wedge *(u_h \wedge u_v). \end{aligned}$$

Thus (4.7) follows. $\square$

Lemma 4.6

Let $u=f \,u_h\wedge u_v$ be a smooth degree (a|b) form, where $u_h$ (resp. $u_v$) is a finite wedge product of $\varphi _h^k$ (resp. $\varphi _v^l$) and f is a smooth function. Then

$$\begin{aligned} (d^v)^* u=(-\,1)^a u_h \wedge \left( (-\,1)^{r(b+1)+1}*_v d^v*_v (fu_v) \right) , \end{aligned}$$

(4.8)

and

$$\begin{aligned} (d^h)^*u = \left( (-\,1)^{(n-r)(a+1)+1}*_h d^h*_h (fu_h)\right) \wedge u_v. \end{aligned}$$

(4.9)

Proof

The main idea is to use the fact that $(d^v)^* u$ (resp. $(d^h)^*u$) is the degree $(a|b-1)$ (resp. $(a-1|b)$) part of $d^* u$ and $d^*u=(-\,1)^{n(a+b+1)+1} *d*u$. Thus, the above proposition applies. We shall only prove the first formula. By (4.7), we have

$$\begin{aligned} *u=(-\,1)^{(n-r-a)b} f \, *_h u_h \wedge *_v u_v. \end{aligned}$$

Thus

$$\begin{aligned} d*u=(-\,1)^{(n-r-a)b} \left( df \wedge (*_h u_h \wedge *_v u_v) + f \, d( *_h u_h \wedge *_v u_v) \right) . \end{aligned}$$

Using (4.7) again, we know that the degree $(a|b-1)$-part of $(-1)^{n(a+b+1)+1} *d*u$ is equal to the right hand side of (4.8). $\square$

Remark

Since

$$\begin{aligned} (-1)^{r(b+1)}*_v(\varphi _v^j \wedge )*_v u_v=X_j^v \, \rfloor \, u_v, \end{aligned}$$

(4.8) gives the following formula:

Lemma 4.7

$(d^v)^*=-\,\sum _{j=1}^r (X_j^v \, \rfloor \, ) \wedge (X_j^v).$

Now we can prove the second identity in (4.3).

Lemma 4.8

$[d^h, (d^v)^*]=0$.

Proof

Notice that $d^2\varphi _h^j=0$ gives $d^v C_{kl}^j\equiv 0$. Thus Lemma 4.3 and the above lemma give

$$\begin{aligned} {[}d^h, (d^v)^*]=\sum ([X_j^h, X_k^v])(X_k^v \, \rfloor \, ) \wedge (\varphi _h^j \wedge ). \end{aligned}$$

(4.10)

But by (2) in Definition 3.4, we have

$$\begin{aligned} {[}X_j^h, X_k^v]=0. \end{aligned}$$

(4.11)

Thus the lemma follows. $\square$

The proof of the real case is complete.

4.3 Proof of the complex case

Let $(\mathcal {V}, J, g_X)$ be a complex nilpotent foliation structure on X. Let $\{V_j\}_{1\le j\le n}$ be the local frame of $(T^{1,0}_X, g_X)$ in Definition 3.5). Put

$$\begin{aligned} X_j^h:=V_{r+j}, \ \ X_k^v=V_k, \qquad 1\le k\le r, \ \ 1\le j\le n-s. \end{aligned}$$

We know each $X_k^v$ is vertical and each $X_j^h$ is horizontal. Denote by

$$\begin{aligned} \{\varphi ^j_h, \varphi ^k_v\}_{1\le k\le r, 1\le j\le n-r}, \end{aligned}$$

the dual frame of $\{X_j^h, X_k^v\}$. By (2) in Definition 3.5 and (3.1), we have

$$\begin{aligned} d\varphi ^j_h=\sum _{k,l=1}^{n-r} C_{kl}^j \, \varphi ^k_h \wedge \varphi ^l_h + \sum _{k,l=1}^{n-r} C_{k\bar{l}}^j \, \varphi ^k_h \wedge \overline{\varphi ^l_h}, \ \ 1\le j\le n-r, \end{aligned}$$

and

$$\begin{aligned} d\varphi ^k_v=\sum _{j,l=1}^{n-r} D_{jl}^k \, \varphi ^j_h \wedge \varphi ^l_h + \sum _{j,l=1}^{n-r} D_{j\bar{l}}^k \, \varphi ^j_h \wedge \overline{\varphi ^l_h}, \ \ 1\le k\le r, \end{aligned}$$

where $C_{kl}^j, C_{k\bar{l}}^j, D_{jl}^k$ and $D_{j\bar{l}}^k$ are smooth functions, which gives

Lemma 4.9

The components $\overline{\partial }^h, \overline{\partial }^v, R_{\overline{\partial }}$ can be written as

$$\begin{aligned} \overline{\partial }^h= & {} \sum _{j=1}^{n-r} \overline{\varphi _h^j} \wedge (\overline{X_j^h})+\sum _{j,k,l=1}^{n-r} \overline{C_{kl}^j \, \left( \varphi ^k_h \wedge \varphi ^l_h\right) \wedge (X_j^h\, \rfloor \,)} +\sum _{j,k,l=1}^{n-r} C_{k\bar{l}}^j \, \left( \varphi ^k_h \wedge \overline{\varphi ^l_h}\right) \wedge (X_j^h\, \rfloor \,),\\ \overline{\partial }^v= & {} \sum _{k=1}^{r} \overline{\varphi _v^k} \wedge (\overline{X_k^v}), \end{aligned}$$

and $R_{\overline{\partial }}=R_{KS}+R_{A_1}+R_{A_2}$ satisfies

$$\begin{aligned} R_{KS}=0, \ \ R_{A_1}=\sum _{k=1}^r \sum _{j,l=1}^{n-r} \left( D_{j\bar{l}}^k \, \varphi ^j_h \wedge \overline{\varphi ^l_h}\right) \wedge (X_k^v \,\rfloor \,), \ \ R_{A_2}=\sum _{k=1}^r \sum _{j,l=1}^{n-r} \overline{\left( D_{jl}^k \, \varphi ^j_h \wedge \varphi ^l_h\right) \wedge (X_k^v \,\rfloor \,)}. \end{aligned}$$

By a similar proof as the real case, we have

$$\begin{aligned} {[}\overline{\partial }^v, R_{\overline{\partial }}^*]=0, \end{aligned}$$

(4.12)

and the following analogy of Lemma 4.7.

Lemma 4.10

$(\overline{\partial }^v)^*=-\,\sum _{j=1}^r (\overline{X_j^v} \, \rfloor \, ) \wedge (X_j^v).$

Similar as the real case, the above lemma gives

$$\begin{aligned} {[}\overline{\partial }^h, (\overline{\partial }^v)^*]=0 \end{aligned}$$

(4.13)

We know that (4.12) and (4.13) together give

$$\begin{aligned} \Box _{\overline{\partial }}=\Box _{{\overline{\partial }}^v}+\Box _{{\overline{\partial }}^h+R_{\overline{\partial }}}. \end{aligned}$$

Now it suffices to prove

$$\begin{aligned} \Box _{\overline{\partial }^v}=\Box _{\partial ^v}. \end{aligned}$$

By Lemmas 4.10 and 4.9, we have

$$\begin{aligned} \Box _{\overline{\partial }^v}=-\,\sum _{j=1}^r (X_j^v)(\overline{X_j^v})=-\,\sum _{j=1}^r (\overline{X_j^v})(X_j^v), \end{aligned}$$

(4.14)

which gives

$$\begin{aligned} \Box _{\overline{\partial }^v}=\overline{\Box _{\overline{\partial }^v}}=\Box _{\partial ^v}. \end{aligned}$$

The proof of Theorem 4.1 is complete. $\square$

Remark

One may also prove the complex case of Theorem 4.1 by using vertical–horizontal decomposition of the following Demailly–Griffiths–Kähler identity (see page 306 in [12, 15] or [27] for a pure algebraic proof)

$$\begin{aligned} \overline{\partial }^*=i[\partial , \Lambda ]+[L, \theta ^*], \ \ \ \ \ Lu:=\omega \wedge u,\ \ \theta :=[\overline{\partial }, L], \ \ \Lambda :=L^*, \end{aligned}$$

(4.15)

where $\omega$ denotes the real Hermitian (1, 1)-form associated to $(g_X, J)$.

5 An example: the Kodaira–Thurston manifold

The Kodaira–Thurston surface was first found by Kodaira [17]. It is the first example [28] of complex symplectic manifold without Kähler structure. Let us recall its definition in [27]. Consider the following group structure

$$\begin{aligned} a*b:=(a^1+b^1, a^2+b^2, a^3+a^1b^2+b^3, a^4+b^4), \end{aligned}$$

on $\mathbb {R}^4$. The Kodaira–Thurston surface X is defined as the quotient manifold with respect to the left action of $\mathbb {Z}^4$ on (notice that $\mathbb {Z}^4$ is a discrete subgroup of G)

$$\begin{aligned} G:=(\mathbb {R}^4, * ). \end{aligned}$$

It is easy to see that X is a compact manifold with respect to the quotient topology. Let $(x^1, x^2, x^3, x^4)$ be the canonical coordinate system on $\mathbb {R}^4$. We know

$$\begin{aligned} {\left\{ \begin{array}{ll} \varphi _h=\hbox {d}x^1+i\hbox {d}x^2,\\ \varphi _v=\hbox {d}x^3-x^1\hbox {d}x^2+i\hbox {d}x^4,\\ \overline{\varphi _h}=\hbox {d}x^1-i\hbox {d}x^2,\\ \overline{\varphi _v}=\hbox {d}x^3-x^1\hbox {d}x^2-i\hbox {d}x^4, \end{array}\right. } \end{aligned}$$

(5.1)

is a frame of the space of G-invariant (with respect to the left action of G) 1-forms on X. Let J be the almost complex structure on X such that the associated $\wedge ^{1,0}T^{*}X$ is spanned by $\{\varphi _h, \varphi _v\}$. Notice that

$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}\varphi _h=0,\\ \hbox {d}\varphi _v=-\,\frac{i}{2}\varphi _h\wedge \overline{\varphi _h}. \end{array}\right. } \end{aligned}$$

(5.2)

implies that J is integrable on X. One may check that

$$\begin{aligned} {\left\{ \begin{array}{ll} \overline{\partial }(x^1+ix^2)=0,\\ \overline{\partial }\left( x^3+ix^4-\frac{i}{2}(x^1)^2\right) =0. \end{array}\right. } \end{aligned}$$

Thus

$$\begin{aligned} {\left\{ \begin{array}{ll} z:=x^1+ix^2,\\ w:=x^3+ix^4-\frac{i}{2}(x^1)^2. \end{array}\right. } \end{aligned}$$

are local holomorphic coordinates on X. Now we know that the following holomorphic map from $(\mathbb {R}^4, J)$ to $\mathbb {C}$

$$\begin{aligned} (z, w)\mapsto z, \end{aligned}$$

defines a holomorphic submersion, say $\pi$, from X to the torus $\mathbb {T}:=\mathbb {C}/ \mathbb {Z}^2$. Let $\{X^h, X^v\}$ be the global frame of $T^{1,0}(X)$ that is dual to $\{\varphi _h, \varphi _v\}$. Then we know that

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial /\partial z:=X^h+ix^1 X^v,\\ \partial /\partial w:=X^v. \end{array}\right. } \end{aligned}$$

is a holomorphic $\pi$-local (i.e., well defined on the $\pi$-inverse of a sufficiently small open set in $\mathbb {T}$) frame for $T^{1,0}X$. Thus we have

Proposition 5.1

$\pi : X\rightarrow \mathbb {T}$ is locally trivial.

Remark

Notice that (5.2) implies that the fibers of $\pi$ defines a complex nilpotent foliation structure on $(X, J, g_X)$, where $g_X$ is J-hermitian such that the fundamental form of $(g_X, J)$ is

$$\begin{aligned} \omega =i \varphi _h\wedge \overline{\varphi _h}+ i \varphi _v\wedge \overline{\varphi _v}. \end{aligned}$$

We know that $\varphi _h,\overline{\varphi _h}$ are horizontal forms and $\varphi _v, \overline{\varphi _v}$ are vertical forms. By (5.2), we know that

$$\begin{aligned} R_{\overline{\partial }}=-\,\frac{i}{2}\varphi _h\wedge \overline{\varphi _h}\wedge (X^v\,\rfloor \,\ldots ), \end{aligned}$$

is of degree $(1,1|-1,0)$. We shall use Theorem 4.1 and Corollary 4.2 to give another proof of the following well known result (see Sect. 5 in [9]).

Theorem 5.2

Denote by $\mathcal {H}^{p,q}(\overline{\partial })$ the space of $\overline{\partial }$-harmonic (p, q)-forms on the Kodaira–Thurston surface $(X, J, g_X)$, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {H}^{0,0}(\overline{\partial }) = \mathrm{Span}_{\mathbb {C}}\,\langle 1\rangle ,\\ \mathcal {H}^{1,0}(\overline{\partial })= \mathrm{Span}_{\mathbb {C}}\,\langle \varphi _h\rangle , \\ \mathcal {H}^{0,1}(\overline{\partial }) = \mathrm{Span}_{\mathbb {C}}\,\langle \overline{\varphi _h},\ \overline{\varphi _v}\rangle , \\ \mathcal {H}^{2,0}(\overline{\partial }) = \mathrm{Span}_{\mathbb {C}}\,\langle \varphi _h \wedge \varphi _v \rangle ,\\ \mathcal {H}^{1,1}(\overline{\partial }) = \mathrm{Span}_{\mathbb {C}}\,\langle \overline{\varphi _h} \wedge \varphi _v,\ \varphi _h\wedge \overline{\varphi _v}\rangle , \\ \mathcal {H}^{0,2}(\overline{\partial }) = \mathrm{Span}_{\mathbb {C}}\,\langle \overline{\varphi _h} \wedge \overline{\varphi _v}\rangle , \\ \mathcal {H}^{2,1}(\overline{\partial }) = \mathrm{Span}_{\mathbb {C}}\,\langle \varphi _h \wedge \varphi _v \wedge \overline{\varphi _h}, \ \varphi _h \wedge \varphi _v \wedge \overline{\varphi _v}\rangle ,\\ \mathcal {H}^{1,2}(\overline{\partial }) = \mathrm{Span}_{\mathbb {C}}\,\langle \varphi _v\wedge \overline{\varphi _h} \wedge \overline{\varphi _v} \rangle ,\\ \mathcal {H}^{2,2}(\overline{\partial }) = \mathrm{Span}_{\mathbb {C}}\,\langle \varphi _h \wedge \varphi _v\wedge \overline{\varphi _h} \wedge \overline{\varphi _v} \rangle . \end{array}\right. } \end{aligned}$$

Proof

$\mathcal {H}^{0,0}(\overline{\partial }) = \mathrm{Span}_{\mathbb {C}}\,\langle 1\rangle$ is trivial. By Corollary 4.2, we know that all harmonic forms in $\mathcal {H}^{p,q}(\overline{\partial })$ lie in the kernel of $\overline{\partial }^v$, $(\overline{\partial }^v)^*$, $\overline{\partial }^h+R_{\overline{\partial }}$ and $(\overline{\partial }^h+R_{\overline{\partial }})^*$.

Degree (1, 0) case: Let

$$\begin{aligned} u=a \varphi _h+ b\varphi _v, \end{aligned}$$

be in $\mathcal {H}^{1,0}(\overline{\partial })$. Notice that $\overline{\partial }^v u=0$ is equivalent to

$$\begin{aligned} \overline{\partial }^v a=\overline{\partial }^v b=0, \end{aligned}$$

and $(\overline{\partial }^h+R_{\overline{\partial }})u=0$ is equivalent to

$$\begin{aligned} \overline{\partial }^h b=0, \ \ \overline{\partial }^h a+\frac{i}{2} b\, \overline{\varphi _h}=0. \end{aligned}$$

Thus $\overline{\partial }b =0$ and b is a constant. Notice that $\overline{\partial }^v a=0$ and $\overline{\partial }^h a+\frac{i}{2} b\, \overline{\varphi _h}=0$ together imply $\frac{i}{2} b\, \overline{\varphi _h}=-\,\overline{\partial }a$ is $\overline{\partial }$-exact. Since $\overline{\varphi _h}$ is not $\overline{\partial }$-exact, we know that $b=0$. Thus

$$\begin{aligned} \overline{\partial }^h a=\overline{\partial }^v a=0, \end{aligned}$$

which gives $\overline{\partial }a =0$ and a is a constant. Thus $\mathcal {H}^{1,0}(\overline{\partial })= \mathrm{Span}_{\mathbb {C}}\,\langle \varphi _h\rangle$.

Degree (0, 1) case: Let

$$\begin{aligned} u=a \overline{\varphi _h}+ b\overline{\varphi _v}, \end{aligned}$$

be in $\mathcal {H}^{0,1}(\overline{\partial })$. $\overline{\partial }^v u=(\overline{\partial }^h+R_{\overline{\partial }}) u=0$ is equivalent to

$$\begin{aligned} \overline{\partial }^v a=\overline{\partial }^h b=0. \end{aligned}$$

$(\overline{\partial }^v)^*u=0$ is equivalent to

$$\begin{aligned} \partial ^v b=0. \end{aligned}$$

Since $\Box _{\overline{\partial }^v}=\Box _{\partial ^v}$, we know that $\partial ^v b=0$ implies $\overline{\partial }^v b=0$. Thus $\overline{\partial }b=0$ and b is a constant. $(\overline{\partial }^h+R_{\overline{\partial }})^*u=0$ is equivalent to $\partial ^h a=0$, thus $\partial a=0$ and a is a constant.

Degree (2, 0) case: Let

$$\begin{aligned} u=a \varphi _h\wedge \varphi _v, \end{aligned}$$

be in $\mathcal {H}^{2,0}(\overline{\partial })$. $\overline{\partial }u=0$ is equivalent to $\overline{\partial }a=0$, which is equivalent to that a is a constant.

Degree (1, 1) case: Let

$$\begin{aligned} u=a \varphi _h\wedge \overline{\varphi _h} +b \varphi _h\wedge \overline{\varphi _v} +c \varphi _v\wedge \overline{\varphi _h} +f \varphi _v\wedge \overline{\varphi _v} , \end{aligned}$$

be a harmonic (1, 1)-form. We have

$$\begin{aligned} \Box _{\overline{\partial }^v}u= (\Box _{\overline{\partial }^v}a)\, \varphi _h\wedge \overline{\varphi _h} + (\Box _{\overline{\partial }^v}b) \, \varphi _h\wedge \overline{\varphi _v} + (\Box _{\overline{\partial }^v}c )\, \varphi _v\wedge \overline{\varphi _h} + (\Box _{\overline{\partial }^v}f)\, \varphi _v\wedge \overline{\varphi _v}, \end{aligned}$$

Then $\Box _{\overline{\partial }^v}u =0$ is equivalent to

$$\begin{aligned} \Box _{\overline{\partial }^v}a=\Box _{\overline{\partial }^v}b =\Box _{\overline{\partial }^v}c=\Box _{\overline{\partial }^v}f=0. \end{aligned}$$

Together with $\Box _{\partial ^v}=\Box _{\overline{\partial }^v}$, the above identities give

$$\begin{aligned} d^v a=d^v b=d^v c= d^v f=0. \end{aligned}$$

$(\overline{\partial }^h+R_{\overline{\partial }})u=0$ is equivalent to

$$\begin{aligned} \overline{\partial }^h f=0, \qquad \overline{\partial }^h b+\frac{i}{2} f \overline{\varphi _h}=0, \end{aligned}$$

thus $\overline{\partial }f=0$ and f is a constant. Since $\overline{\partial }^v b=0$, we have

$$\begin{aligned} \overline{\partial }b+\frac{i}{2} f \overline{\varphi _h}=0. \end{aligned}$$

By the above computation of $\mathcal {H}^{0,1}(\overline{\partial })$, we know that $\overline{\varphi _h}$ is not $\overline{\partial }$-exact. Thus $f=0$ and $\overline{\partial }b=0$. Now we know that b is a constant. $(\overline{\partial }^h+R_{\overline{\partial }})^*u=0$ is equivalent to

$$\begin{aligned} \partial ^h a=0, \qquad \frac{i}{2} a \varphi _h+\partial ^h c=0. \end{aligned}$$

thus $\partial a=0$ and a is a constant. Again

$$\begin{aligned} -\,\frac{i}{2} \bar{a} \overline{\varphi _h}+\overline{\partial }\bar{c}=0, \end{aligned}$$

gives $a=0$ and c is a constant.

For the remaining cases, by the following well-known formula

$$\begin{aligned} \dim \mathcal {H}^{p,q}(\overline{\partial })=\dim \mathcal {H}^{n-p,n-q}(\overline{\partial }), \end{aligned}$$

it is enough to check that the listed forms lie in the$\overline{\partial }$-harmonic spaces, which follows by a direct computation. $\square$

5.1 Nilpotent fibrations

For the Kodaira–Thurston manifold, the complex nilpotent foliation structure comes from a holomorphic fibration. The general definition is as follows:

Definition 5.3

(Nilpotent fibration) We call a proper smooth submersion $\pi : (X, g_X)\rightarrow (B, g_B)$ between two Riemannian manifolds a nilpotent fibration if the associated foliation $\mathcal {V}$ of the fibers defines a nilpotent foliation structure on $(X, g_X)$ and

$$\begin{aligned} g_X(V,W)=g_B(\pi _*V, \pi _* W), \end{aligned}$$

(5.3)

for every horizontal vector fields V, W on X.

Remark

Let $V_B$ be a vector field on B, we call a vector field $V_X$ on X a lift of $V_B$ if

$$\begin{aligned} \pi _* V_X=V_B. \end{aligned}$$

It is clear that $V_B$ has a unique lift $V_X$ such that $V_X$ is horizontal. (5.3) says that the norm of a vector field on B is equal to the norm of its horizontal lift.

Definition 5.4

(Complex nilpotent fibration) We call a proper holomorphic submersion $\pi : (X, \omega _X)\rightarrow (B, \omega _B)$ between two Hermitian complex manifolds a complex nilpotent fibration if the associated foliation $\mathcal {V}$ of the fibers defines a complex nilpotent foliation structure on $(X, \omega _X)$ and

$$\begin{aligned} \omega _X(V,W)=\omega _B(\pi _*V, \pi _* W), \end{aligned}$$

(5.4)

for every horizontal (1, 0)-vector fields V, W on X.

Theorem 4.1 gives

Theorem 5.5

On the total space of a nilpotent fibration, we have

$$\begin{aligned} \Box _d=\Box _{d^v}+\Box _{d^h+R_d}. \end{aligned}$$

On the total space of a complex nilpotent fibration, we have

$$\begin{aligned} \Box _{\overline{\partial }}=\Box _{{\overline{\partial }}^v}+\Box _{{\overline{\partial }}^h+R_{\overline{\partial }}}, \ \Box _{\overline{\partial }^v}=\Box _{\partial ^v}. \end{aligned}$$

Remark

Associated to a fibration there is a natural Leray-Serre spectral sequence, which plays a crucial role in the proof of Nomizu-type theorems. But in general a foliation does not give a good fibration structure, thus one has to use other methods. Our main idea is: with the help of Theorem 4.1, one may use the spectral sequence for double complex to continue the reduction process (as in the fibration case), in which the natural setup is a manifold with a nilpotent frame.

6 Nilpotent frame

6.1 Real case

Let X be a compact smooth manifold with trivial $T^*_X$. Let

$$\begin{aligned} \Phi :=\{\sigma ^1, \ldots , \sigma ^n\} \end{aligned}$$

be a global smooth frame of $T^*_X$. Then we have the following Maurer–Cartan equations

$$\begin{aligned} \hbox {d}\sigma ^j =\sum _{k,l=1}^n A_{kl}^j \, \sigma ^k \wedge \sigma ^l, \end{aligned}$$

(6.1)

where $A_{kl}^j$ are globally defined smooth functions on X. Recall that $\Phi$ is a nilpotent frame if $A_{kl}^j$ are real constants and the above equations reduce to

$$\begin{aligned} \hbox {d}\sigma ^j =\sum _{k,l>j} A_{kl}^j \, \sigma ^k \wedge \sigma ^l. \end{aligned}$$

(6.2)

Definition 6.1

Let $\Phi$ be a nilpotent frame. Put $r_0=0$ and define $r_j$ ($j\ge 1$) inductively by

$$\begin{aligned} r_j+1=\min U_j , \ U_j:=\cup _{j\ge r_{j-1}+1} \{k, l : A_{kl}^j \ne 0\}, \end{aligned}$$

where $r_j:=n$ if $U_j$ is empty. Fix k such that

$$\begin{aligned} 0=r_0<r_1< \cdots< r_{k-1}<r_k=n, \end{aligned}$$

we call $\Phi$ a k-nilpotent frame. Let $S^j$ be the subbundle of $T^*_X$ generated by $\{\sigma ^1,\ldots , \sigma ^{r_j}\}$. We call

$$\begin{aligned} 0=S^0 \hookrightarrow S^1 \hookrightarrow \cdots \hookrightarrow S^k=T^*_X, \end{aligned}$$

the $\Phi$-filtration of $T^*_X$.

Remark

We always have $1\le k\le n$. For the Kodaira–Thurston manifold (see (5.1)), put

$$\begin{aligned} \sigma ^1=\hbox {d}x^4, \ \sigma ^2=\hbox {d}x^3-x^1\hbox {d}x^2, \ \sigma ^3=\hbox {d}x^2, \ \sigma ^4=\hbox {d}x^1, \end{aligned}$$

we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}\sigma ^1=0,\\ \hbox {d}\sigma ^2=\sigma ^3\wedge \sigma ^4,\\ \hbox {d}\sigma ^3=0,\\ \hbox {d}\sigma ^4=0. \end{array}\right. } \end{aligned}$$

Thus $k=2$ and the $\Phi$-filtration is

$$\begin{aligned} 0 \hookrightarrow \mathrm{Span} \{\sigma ^1,\sigma ^2\} \hookrightarrow \mathrm{Span} \{\sigma ^1,\sigma ^2, \sigma ^3,\sigma ^4\} =T^*_X. \end{aligned}$$

Put

$$\begin{aligned} T^*_{v}= \mathrm{Span} \{\sigma ^1,\sigma ^2\}, \ T^*_{h}= \mathrm{Span} \{\sigma ^3,\sigma ^4\}. \end{aligned}$$

We get the following vertical–horizontal decomposition of the $T^*_X$

$$\begin{aligned} T^*_X=T^*_{v} \oplus T^*_{h}. \end{aligned}$$

with respect the Riemannian metric $\sum \sigma ^j \otimes \sigma ^j$. In general, we shall introduce the following definition

Definition 6.2

Denote by $T^*_{v_j}$ ($1\le j\le k$) the subbundle of $T^*_X$ generated by $\{\sigma ^{r_{j-1}+1}, \ldots , \sigma ^{r_j}\}$. Put

$$\begin{aligned} T^*_{h_j}= \oplus _{l=j+1}^k T^*_{v_l}, \ 0\le j\le k-1, \ \ T^*_{h_k}=0. \end{aligned}$$

We call

$$\begin{aligned} T^*_{h_{j-1}}:=T^*_{v_{j}}\oplus T^*_{h_{j}}, \ \ 1\le j\le k, \end{aligned}$$

the jthvertical horizontal decomposition with respect to the following Riemannian metric

$$\begin{aligned} g_X:=\sum _{j=1}^n \sigma ^j \otimes \sigma ^j, \end{aligned}$$

(6.3)

on X associated to the nilpotent frame $\Phi$.

Remark

The first vertical horizontal decomposition of $T^*X$ gives

$$\begin{aligned} d:=d^{h_0}=d^{v_1}+d^{h_1}+R_d^1, \end{aligned}$$

where $d^{v_1}$ (resp. $d^{h_1})$ increases the first vertical (resp. horizontal) degree by one and $R_d^1$ is the remaining term. In general, the jth vertical horizontal decomposition gives

$$\begin{aligned} d^{h_{j-1}}=d^{v_j}+d^{h_j}+R_d^j, \qquad 1\le j\le k. \end{aligned}$$

Notice that the kth vertical horizontal decomposition reduces to

$$\begin{aligned} d^{h_{k-1}}=d^{v_k}, \ \ R_d^k=0, \ d^{h_k}=0. \end{aligned}$$

We shall use our fundamental theorem (see Theorem 4.1) to prove the following result.

Theorem 6.3

With respect to the notation above, we have

$$\begin{aligned} \Box _d=\Box _{d^{v_1}}+\Box _{d^{h_1}+ R_d^1}, \end{aligned}$$

(6.4)

on the space of smooth forms on X. Moreover, if $k\ge 2$ then for every $2\le j \le k$, we have

$$\begin{aligned} \Box _{d^{h_{j-1}}}= \Box _{d^{v_{j}}}+\Box _{d^{h_{j}}+ R_d^{j}}, \end{aligned}$$

(6.5)

on the space of smooth forms in $\ker \Box _{d^{v_1}} \cap \cdots \cap \ker \Box _{d^{v_{j-1}}}$.

Proof

Denote by $\{T^{h_j}, T^{v_k}\}$ the subbundles of $T_X$ that are dual to $\{T^*_{h_j}, T^*_{v_k}\}$. From Definition 6.1, we know that the distribution $\mathcal {V}^1$ associated to $T^{v_1}$ is integrable and $\mathcal {V}^1$ defines a nilpotent foliation with respect to the Riemannian metric $g_X$ in (6.3). Thus Theorem 4.1 gives (6.4).

Now let us prove (6.5) for $j=2$. Let us write a smooth form u on X as

$$\begin{aligned} u=\sum f_{p,q} \,u_{h_1}^p \wedge u_{v_1}^q, \end{aligned}$$

(6.6)

where $\{u_{h_1}^p\}$ (resp. $\{u_{v_1}^q\}$) denotes a basis of the exterior algebra generated by $\{\sigma ^{r_1+1}, \ldots , \sigma ^{n}\}$ (resp. $\{\sigma ^1, \ldots , \sigma ^{r_1}\}$), respectively. Denote by $\{V_l\}$ the frame of $T_X$ that is dual to $\{\sigma ^l\}$. Put

$$\begin{aligned} \varphi ^l_{v_1}=\sigma ^l, \ \ \varphi ^m_{h_1}=\sigma ^{r_1+m}, \ \ \ \ X_l^{v_1}=V_l, \ \ X_m^{h_1}=X_{r_1+m}. \end{aligned}$$

By Lemmas 4.3 and 4.10, we have

$$\begin{aligned} \Box _{d^{v_1}} =-\,\sum (X^{v_1}_l) (X^{v_1}_l), \end{aligned}$$

which gives

$$\begin{aligned} \Box _{d^{v_1}} u= \sum (\Box _{d^{v_1}} f_{p,q}) \,u_{h_1}^p \wedge u_{v_1}^q. \end{aligned}$$

Thus $\Box _{d^{v_1}} u=0$ is equivalent to

$$\begin{aligned} \Box _{d^{v_1}} f_{p,q}\equiv 0, \end{aligned}$$

for all p, q. Since $f_{p,q}$ are globally defined smooth functions on X, we know that $\Box _{d^{v_1}} f_{p,q}\equiv 0$ is equivalent to that $d^{v_1} f_{p,q}\equiv 0$. Thus we get the following lemma.

Lemma 6.4

$\Box _{d^{v_1}} u=0$ is equivalent to that all $f_{p,q}$ are constants on leaves of $\mathcal {V}^1$.

By the above lemma and Definition 6.1, we know that the distribution $\mathcal {V}^2$ associated to $T^{v_2}$ is integrable on the space of smooth forms in $\ker \Box _{d^{v_1}}$. Moreover, on the space of smooth forms in $\ker \Box _{d^{v_1}}$, $\mathcal {V}^2$ defines a nilpotent foliation with respect to the following decomposition

$$\begin{aligned} T^*_{h_1}=T^*_{v_2}\oplus T^*_{h_2}. \end{aligned}$$

Thus the proof of Theorem 4.1 gives (6.5) for $j=2$. The general case follows by induction on j. $\square$

6.2 Complex case

Let X be a compact complex manifold with smoothly trivial holomorphic cotangent bundle $\wedge ^{1,0}T^*_X$. Let

$$\begin{aligned} \Psi :=\{\xi ^1, \ldots , \xi ^n\} \end{aligned}$$

be a global smooth frame of $\wedge ^{1,0}T^*_X$. Recall that $\Psi$ is a complex nilpotent frame if

$$\begin{aligned} d\xi ^j =\sum _{k,l>j} B_{kl}^j \, \xi ^k \wedge \xi ^l+\sum _{k,l>j} B_{k\bar{l}}^j \, \xi ^k \wedge \overline{\xi ^l}, \end{aligned}$$

(6.7)

where $B_{kl}^j$ and $B_{k\bar{l}}^j$ are complex constants.

Definition 6.5

Let $\Psi$ be a complex nilpotent frame. Put $r_0=0$ and define $r_j$ ($j\ge 1$) inductively by

$$\begin{aligned} r_j+1=\min U_j , \ U_j:=\cup _{j\ge r_{j-1}+1} \{k, l : B_{kl}^j \ \text {or} \ B_{k\bar{l}}^j \ne 0\}, \end{aligned}$$

where $r_j:=n$ if $U_j$ is empty. Fix k such that

$$\begin{aligned} 0=r_0<r_1< \cdots< r_{k-1}<r_k=n, \end{aligned}$$

we call $\Psi$ a complex k-nilpotent frame. Let $S^j$ be the subbundle of $\wedge ^{1,0}T^*_X$ generated by $\{\xi ^1,\ldots , \xi ^{r_j}\}$. We call

$$\begin{aligned} 0=S^0 \hookrightarrow S^1 \hookrightarrow \cdots \hookrightarrow S^k=\wedge ^{1,0} T^*_X, \end{aligned}$$

the $\Psi$-filtration of $\wedge ^{1,0}T^*_X$.

Similar as the real case, we have

Definition 6.6

Denote by $T^*_{v_j}$ ($1\le j\le k$) the (smooth, may not be holomorphic) subbundle of $\wedge ^{1,0} T^*_X$ generated by $\{\xi ^{r_{j-1}+1}, \ldots , \xi ^{r_j}\}$. Put

$$\begin{aligned} T^*_{h_j}= \oplus _{l=j+1}^k T^*_{v_l}, \ 0\le j\le k-1, \ \ T^*_{h_k}=0. \end{aligned}$$

We call

$$\begin{aligned} T^*_{h_{j-1}}:=T^*_{v_{j}}\oplus T^*_{h_{j}}, \ \ 1\le j\le k, \end{aligned}$$

the jth vertical horizontal decomposition with respect to the following Hermitian form

$$\begin{aligned} \omega _X:=i\, \sum _{j=1}^n \xi ^j \wedge \overline{\xi ^j}, \end{aligned}$$

(6.8)

on X associated to the complex nilpotent frame $\Psi$.

Remark

The first vertical horizontal decomposition of $T^*X$ gives

$$\begin{aligned} \overline{\partial }:=\overline{\partial }^{h_0}=\overline{\partial }^{v_1}+\overline{\partial }^{h_1}+R_{\overline{\partial }}^1, \end{aligned}$$

where $\overline{\partial }^{v_1}$ (resp. $\overline{\partial }^{h_1})$ increases the first vertical (resp. horizontal) degree by (0, 1) and $R_{\overline{\partial }}^1$ is the remaining term. In general, the jth vertical horizontal decomposition gives

$$\begin{aligned} \overline{\partial }^{h_{j-1}}=\overline{\partial }^{v_j}+\overline{\partial }^{h_j}+R_{\overline{\partial }}^j, \qquad 1\le j\le k. \end{aligned}$$

Notice that the kth vertical horizontal decomposition reduces to

$$\begin{aligned} \overline{\partial }^{h_{k-1}}=\overline{\partial }^{v_k}, \ \ R_{\overline{\partial }}^k=0, \ \overline{\partial }^{h_k}=0. \end{aligned}$$

Similar as the real case, Theorem 4.1 implies the following result.

Theorem 6.7

With respect to the notation above, we have

$$\begin{aligned} \Box _{\overline{\partial }}=\Box _{\overline{\partial }^{v_1}}+\Box _{\overline{\partial }^{h_1}+ R_{\overline{\partial }}^1}, \ \ \ \Box _{\overline{\partial }^{v_1}}=\Box _{\partial ^{v_1}}, \end{aligned}$$

(6.9)

on the space of smooth forms on X. Moreover, if $k\ge 2$ then for every $2\le j \le k$, we have

$$\begin{aligned} \Box _{\overline{\partial }^{h_{j-1}}}=\Box _{\overline{\partial }^{v_j}}+\Box _{\overline{\partial }^{h_j}+ R_{\overline{\partial }}^j}, \ \ \ \Box _{\overline{\partial }^{v_j}}=\Box _{\partial ^{v_j}}, \end{aligned}$$

(6.10)

on the space of smooth forms in $\ker \Box _{\overline{\partial }^{v_1}} \cap \cdots \cap \ker \Box _{\overline{\partial }^{v_{j-1}}}$.

7 Proof of the main theorem

7.1 Cohomology description of Theorem 4.1

Let X be a compact smooth manifold. By the de Rham theorem, we have the following isomorphism

$$\begin{aligned} H_d^{\star } \simeq \mathcal {H}_d^{\star }:=\{u\in \mathcal {A}^*: \Box _d u=0\}, \end{aligned}$$

where $H_d^{\star }$ denotes the de Rham cohomology group of X, $\mathcal {A}^*$ denotes the space of smooth forms on X. With the assumption in Theorem 4.1 (real case), every $u\in \mathcal {A}^*$ must satisfies

$$\begin{aligned} \Box _{d^v} u=0, \end{aligned}$$

which gives, by Lemma 6.4, that all coefficients of u are constants on leaves of $\mathcal {V}$. Denote by $\mathcal {A}_1^*$ the space of smooth forms on X whose coefficients are constants on leaves of $\mathcal {V}$, the usual d operator reduces to $d^h+R_d$ on $\mathcal {A}_1^*$, which suggests to look at the following cohomology (notice that $(d^h+R_d)^2=0$ on $\mathcal {A}_1^*$, since the coefficients of $R_d$ are constants on leaves of $\mathcal {V}$)

$$\begin{aligned} H_{d^h+R_d}^*:=\frac{\{u\in \mathcal {A}_1^*: (d^h+R_d) u=0\}}{(d^h+R_d)(\mathcal {A}_1^*)}. \end{aligned}$$

A representative of a class in $H_{d^h+R_d}^*$ is minimal if and only if it lies in $\ker \Box _{d^h+R_d}$. Thus Theorem 4.1 implies that

$$\begin{aligned} \mathcal {H}_d^{\star } \simeq H_{d^h+R_d}^*. \end{aligned}$$

A similar argument also works in the complex case, to summarize, we get

Theorem 7.1

Theorem 4.1 (real case) implies

$$\begin{aligned} H_d^{*} \simeq H_{d^h+R_d}^*. \end{aligned}$$

Theorem 4.1 (real case) implies

$$\begin{aligned} H_{\overline{\partial }}^{*, *} \simeq H_{\overline{\partial }^h+R_{\overline{\partial }}}^{*, *}. \end{aligned}$$

Remark

In the setting of Theorems 6.7 and 6.3, one may continue to get a sequence of isomorphisms. In fact, with the notation in Theorem 6.3, denote by $\mathcal {V}^j$ ($1\le j \le k$) the distribution associated to $T^{v_1}\oplus \cdots \oplus T^{v_j}$, let $\mathcal {A}_j^*$ be the space of smooth forms on X whose coefficients are constants on leaves of $\mathcal {V}^j$, then one may verify that

$$\begin{aligned} (d^{h_j}+R^j_{d})^2=0, \qquad \ \text {on}\ \mathcal {A}_j^*. \end{aligned}$$

Put

$$\begin{aligned} H_{d^{h_j}+R^j_d}^*:=\frac{\{u\in \mathcal {A}_j^*: (d^{h_j}+R^j_{d}) u=0\}}{(d^{h_j}+R^j_{d})(\mathcal {A}_j^*)}, \ \ H_{d^{h_j}}^*:=\frac{\{u\in \mathcal {A}_{j}^*: d^{h_j}u=0\}}{d^{h_j}(\mathcal {A}_j^*)} \end{aligned}$$

we have

Theorem 7.2

Theorem 6.3 implies

$$\begin{aligned} H_d^{*} \simeq H_{d^{h_1}+R^1_{d}}^*, \ \ H_{d^{h_{j-1}}}^* \simeq H_{d^{h_j}+R^j_d}^*, \ \ \forall \ 2\le j\le k. \end{aligned}$$

Theorem 6.7 implies

$$\begin{aligned} H_{\overline{\partial }}^{*, *} \simeq H_{\overline{\partial }^{h_1}+R^1_{\overline{\partial }}}^{*, *}, \ \ H_{\overline{\partial }^{h_{j-1}}}^{*, *} \simeq H_{\overline{\partial }^{h_j}+R^j_{\overline{\partial }}}^{*,*}, \ \ \forall \ 2\le j\le k. \end{aligned}$$

Remark

In the proper fibration case, the above theorem is essentially equivalent to the use of the $E_2$ term in the Leray–Serre spectral sequence.

In order to apply the above theorem, one has to study the relation between $H_{d^{h_j}+R^j_d}^*$ and $H_{d^{h_j}}^*$; in the next section, we shall prove that, with respect to the spectral sequence of the double complex $d^{h_j}+R^j_d$, the $E_2$ term is equal to $H_{d^{h_j}}^*$ and the $E_\infty$ term is $H_{d^{h_j}+R^j_d}^*$.

7.2 Spectral sequence for double complex

We shall use the notation in [14]. We look at the complex

$$\begin{aligned} d^{h_j}+R^j_d: \mathcal {A}_j^* \rightarrow \mathcal {A}_j^*, \end{aligned}$$

denote its kernel by Z, image by B and put

$$\begin{aligned} H:=\frac{Z}{B}=H_{d^{h_j}+R^j_d}^*. \end{aligned}$$

With the notation in [14], we denote by $^p T$ as the space of forms in $\mathcal {A}_j^*$ whose jth horizontal degree is no less than p. By Definition 6.1, the maximal jth horizontal degree is $n-r_j$, thus we have

$$\begin{aligned} \mathcal {A}_j^*\, =\, ^0 T\ \supset \, ^1 T \, \supset \, \cdots \, \supset \, ^{n-r_j} T \, \supset \, ^{n-r_j+1} T\,=\, \{0\}. \end{aligned}$$

Denote by $T^m$ the space of degree-m forms in $\mathcal {A}_j^*$. We have

$$\begin{aligned} \mathcal {A}_j^*=\oplus _{m=0}^n T^m. \end{aligned}$$

Put

$$\begin{aligned} ^p T^m:\,=\, ^p T \cap T^m, \end{aligned}$$

we know that $^p T^m$ is the space of degree-m forms in $\mathcal {A}_j^*$ whose horizontal degree is no less that p. One may easily verify the following compatibility conditions in [14]

$$\begin{aligned} (d^{h_j}+R^j_d) \,T^m\subset T^{m+1}, \ \ \ (d^{h_j}+R^j_d) \ ^p T \subset \, ^p T. \end{aligned}$$

In fact, notice that $d^{h_j} \, ^p T \subset \, ^{p+1} T$ and $R^j_d \, ^p T \subset \, ^{p+2} T$, the following stronger compatibility condition holds

$$\begin{aligned} (d^{h_j}+R^j_d) \ ^p T \subset \, ^{p+1} T. \end{aligned}$$

Moreover, we shall introduce

$$\begin{aligned} ^p Z^m:= Z\cap \,^p T^n, \ \ ^p B^m:= B\cap \,^p T^n,\ \ ^p H^m:\,=\, \frac{ ^p Z^m}{ ^p B^m}. \end{aligned}$$

The fundamental definition is the following

Definition 7.3

(Spectral sequence of a filtered complex [14]) The modules of spectral sequence are

$$\begin{aligned} E^{p,q}_r:=\frac{^pT^{p+q}_r}{^pT^{p+q}_{r-1}+\, ^pB^{p+q}_{r-1}}, \ \ \ E^{p,q}_\infty :=\frac{^p Z^{p+q}}{^p Z^{p+q}+\, ^pB^{p+q}}, \ \ r=1, 2, \ldots , \end{aligned}$$

where

$$\begin{aligned} ^pT^{m}_r:=\{ u\in \, ^pT^m: (d^{h_j}+R^j_d) u\in \,^{p+r}T\}, \ \ ^p B^{m}_r:=\{ (d^{h_j}+R^j_d) u\in \, ^pT^m: u\in \,^{p-r}T\}. \end{aligned}$$

One may verify that $d^{h_j}+R^j_d$ induces the following $d_r$ complex

$$\begin{aligned} E_r^{p-r, q+r-1}\xrightarrow {d_r} E_r^{p,q} \xrightarrow {d_r} E_r^{p+r, q-r+1}. \end{aligned}$$

As general properties of the spectral sequence, we have

Proposition 7.4

The associated $d_r$ cohomology gives the $E_{r+1}$ term, more precisely

$$\begin{aligned} E_{r+1}^{p,q}\simeq \frac{\ker d_r \cap E_r^{p,q} }{\mathrm{Im}\, d_r \cap E_r^{p,q}}. \end{aligned}$$

Moreover, we have

$$\begin{aligned} E_\infty ^{p,q}\simeq \frac{^p H^{p+q}}{^{p+1} H^{p+q}}, \ \ E_r^{p,q}=E_\infty ^{p,q}, \ \ \forall \ r\ge n-r_j+1. \end{aligned}$$

Remark

For the double complex $d=\partial +\overline{\partial }$ (also called Frölicher spectral sequence), it is well known that [14]

$$\begin{aligned} E_1^{p,q}\simeq H_{\overline{\partial }}^{p,q}. \end{aligned}$$

In our case, we obtain the following

Lemma 7.5

$\oplus _{p+q= m}E_2^{p,q}\simeq H_{d^{h_j}}^{m}$.

Proof

Recall that

$$\begin{aligned} E^{p,q}_2:=\frac{^pT^{p+q}_2}{^pT^{p+q}_{1}+\, ^pB^{p+q}_{1}}. \end{aligned}$$

Let us write $u\in \, ^pT^m$ as

$$\begin{aligned} u=u^{p|m-p}+u^{p+1|m-p-1}+\cdots , \end{aligned}$$

where $u^{\alpha |\beta }$ denotes the component of u with horizontal degree $\alpha$ and vertical degree $\beta$. Notice that $(d^{h_j}+R^j_d) u$ has no horizontal degree $p+1$ component if and only if

$$\begin{aligned} d^{h_j} u^{p|m-p}=0. \end{aligned}$$

Thus u lies in $^pT^{p+q}_2$ if and only if $d^{h_j} u^{p|m-p}=0$. A similar discussion gives that u lies in $^pT^{p+q}_{1}+\, ^pB^{p+q}_{1}$ if and only if $u^{p|m-p}\in \mathrm{Im}\, d^{h_j}$. Thus the (p|q)-component of $H_{d^{h_j}}^{m}$ is isomorphic to $E_2^{p,q}$. Hence the lemma follows. $\square$

7.3 The final proof

Let us introduce the following notation: let D be a complex on $\mathcal {A}_j^*$, denote its associated cohomology by $H^*_{D}$. Recall that $\mathcal {A}_j^*$ means the space of smooth forms on X whose coefficients are constants on leaves of $\mathcal {V}^j$, in particular

$$\begin{aligned} A^*:=\mathcal {A}_k^* \end{aligned}$$

is the finite dimensional real vector space spanned by wedge products of the frame $\Phi$. Assume that D maps $A^*$ to itself. Put

$$\begin{aligned} H^*_{D, \Phi } :=\frac{\ker D|_{A^*}}{\mathrm{Im}\, D|_{A^*}}, \end{aligned}$$

we say that $H^*_{D}$ is simple if it reduces to $H^*_{D, \Phi }$, i.e.,

$$\begin{aligned} H^*_{D} \simeq H^*_{D, \Phi }. \end{aligned}$$

The real case of our main theorem is of course equivalent to that $H_d^*$ is simple. By Theorem 7.2, $H_d^*$ is simple if and only if $H_{d^{h_1}+R^1_{d}}^*$ is simple. Now let us look at the double complex $(d^{h_1}, R^1_{d})$, by Lemma 7.5, its $E_2$ term is $H^*_{d^{h_1}}$. Thus in order to prove that $H_{d^{h_1}+R^1_{d}}^*$ is simple, it suffices to show that $H^*_{d^{h_1}}$ is simple. Apply Theorem 7.2 again, we know that $H^*_{d^{h_1}}$ is simple if and only if $H_{d^{h_2}+R^2_{d}}^*$ is simple. Now again Lemma 7.5 ensures that it is enough to prove simplicity of $H_{d^{h_2}}^*$. Repeat the above argument, we know it suffices to show that $H_{d^{h_k}}^*$ is simple, which holds trivially since $\mathcal {A}_k^*=A^*$. The complex case follows by a similar argument.

Remark

Notice that the Lie algebra cohomology $H_{d, \Phi }^*$ has the following “harmonic” representative

$$\begin{aligned} H_{d, \Phi }^*\simeq \mathcal {H}_{d, \Phi }^*:=\{ u\in A^*: \Box _{R_d^1+\cdots + R_d^{k-1}} u=0\}. \end{aligned}$$

Since

$$\begin{aligned} \mathcal {H}_{d, \Phi }^* \in \ker \Box _{d}, \end{aligned}$$

our main theorem actually also gives a formula for the following d-harmonic space

$$\begin{aligned} \mathcal {H}_{d}^*:=\ker \Box _d=\mathcal {H}_{d, \Phi }^*. \end{aligned}$$

A more careful study of $\mathcal {H}_{d, \Phi }^*$ will be given in Sect. 8.

7.4 Proof of the strong version of Cordero–Fernández–Gray–Ugarte’s theorem

Let us choose an appropriate basis of left-invariant vector fields for the nilpotent Lie group and a left-invariant metric making this a global orthonormal frame. Then we know that our main theorem applies.

Remark

The above argument also applies to the original version of Nomizu’s theorem.

8 An explicit version of our main theorem

In case $k=2$, the remark at the end of Sect. 7.3 gives

$$\begin{aligned} \ker \Box _d=\{ u\in A^*: \Box _{R_d^1} u=0\}. \end{aligned}$$

(8.1)

The above formula can also be proved using the $E_3$ term of the spectral sequence of the double complex $(d^{h_1}, R_d^1)$. In fact, by Lemma 7.5, we know that the $E_3$ term is defined by the following complex

$$\begin{aligned} R_d^1: A^* \rightarrow A^*. \end{aligned}$$

The key point here is that both $R_d^1$ and its adjoint send $A^*$ to itself, from which we know that the above spectral sequence degenerates at $E_3$.

Example 1

By the example in Sect. 5, if X is the Kodaira–Thurston manifold, then $k=2$ and

$$\begin{aligned} R^1_{\overline{\partial }}=-\,\frac{i}{2}\varphi _h\wedge \overline{\varphi _h}\wedge (X^v\,\rfloor \,), \ \ (R^1_{\overline{\partial }})^*=\frac{i}{2}\varphi _v\wedge (\overline{X^h}\,\rfloor \,) (X^h\,\rfloor \,). \end{aligned}$$

The complex version of (8.1) gives

$$\begin{aligned} \mathcal {H}^{*,*}_{\overline{\partial }}=\{ u\in A^{*,*}: R_{\overline{\partial }}^1 u=(R_{\overline{\partial }}^1)^* u=0\}. \end{aligned}$$

Example 2

Let $\mathfrak {g}$ be the 3-dimensional 2-step complex nilpotent Lie algebra admitting a complex (1, 0)-coframe $\{\xi ^1,\xi ^2,\xi ^3\}$ satisfying

$$\begin{aligned} \hbox {d}\xi ^1=\xi ^2\wedge \overline{\xi ^2}+D\xi ^3\wedge \overline{\xi ^3},\qquad \hbox {d}\xi ^2=0,\qquad \hbox {d}\xi ^3=0, \end{aligned}$$

where D is a complex parameter having non-negative imaginary part. Let G be the simply connected Lie group having $\mathfrak {g}$ as Lie algebra. Let X be any compact quotient of G. Then, $k=2$, the decomposition of $\wedge ^{1,0}T^*_X$ into vertical and horizontal subbundles is given by

$$\begin{aligned} \wedge ^{1,0}T^*_X=\langle \xi ^1\rangle \oplus \langle \xi ^2,\xi ^3\rangle \end{aligned}$$

and

$$\begin{aligned} R^1_{\overline{\partial }}=(\xi ^2\wedge \overline{\xi ^2}+D\xi ^3\wedge \overline{\xi ^3})\wedge (\xi _1\,\rfloor \,), \ \ (R^1_{\overline{\partial }})^*=\xi ^1\wedge [(\overline{\xi _2}\,\rfloor \,) (\xi _2\,\rfloor \,)+\bar{D}(\overline{\xi _3}\,\rfloor \,) (\xi _3\,\rfloor \,)], \end{aligned}$$

where $\{\xi _1,\xi _2,\xi _3\}$ satisfies $\xi ^r(\xi _s)=\delta ^r_s$. Also in this case, the complex version of (8.1) gives

$$\begin{aligned} \mathcal {H}^{*, *}_{\overline{\partial }}=\{ u\in A^{*,*}: R_{\overline{\partial }}^1 u=(R_{\overline{\partial }}^1)^* u=0\}. \end{aligned}$$

For general k, we shall introduce the following:

Definition 8.1

With the notation in our main theorem. In the real case, the nilpotent frame $\Phi$ is said to be admissible if $k\ge 2$ or

$$\begin{aligned} R_d^{k-j-1} \left( \Box _{R_d^{k-j}}+\cdots + \Box _{R_d^{k-1}}\right) =\left( \Box _{R_d^{k-j}}+\cdots + \Box _{R_d^{k-1}}\right) R_d^{k-j-1}, \ \ \ \forall \ 1\le j \le k-2, \end{aligned}$$

when $k\ge 3$. In the complex case, the nilpotent frame $\Psi$ is said to be admissible if $k\ge 2$ or

$$\begin{aligned} R_{\overline{\partial }}^{k-j-1} \left( \Box _{R_{\overline{\partial }}^{k-j}}+\cdots + \Box _{R_{\overline{\partial }}^{k-1}}\right) =\left( \Box _{R_{\overline{\partial }}^{k-j}}+\cdots + \Box _{R_{\overline{\partial }}^{k-1}}\right) R_{\overline{\partial }}^{k-j-1}, \ \ \ \forall \ 1\le j \le k-2, \end{aligned}$$

when $k\ge 3$.

We obtain the following result:

Theorem 8.2

With the assumptions in our main theorem. Assume further that the associated frame is admissible. Then:

• (1) In the real case, we have

  $$\begin{aligned} \mathcal {H}_d^{\star } = \{u\in A^{\star }: R_d^j u= (R_d^j)^* u=0, \ \forall \ 1\le j\le k-1\}. \end{aligned}$$

• (2) In the complex case, we have

  $$\begin{aligned} \mathcal {H}_{\overline{\partial }}^{\star , \star } = \{u\in A^{\star , \star }: R_{\overline{\partial }}^j u= (R_{\overline{\partial }}^j)^* u=0, \ \forall \ 1\le j\le k-1\}. \end{aligned}$$

Proof

Notice that

$$\begin{aligned} R_d^{k-2} \left( \Box _{R_d^{k-1}}\right) =\left( \Box _{R_d^{k-1}}\right) R_d^{k-2} \end{aligned}$$

implies (notice that $\mathcal {H}^*_{d^{h_{k-2}}}=\ker \Box _{R_d^{k-1}}$)

$$\begin{aligned} R_d^{k-2} (\mathcal {H}^*_{d^{h_{k-2}}})\subset \mathcal {H}^*_{d^{h_{k-2}}}, \end{aligned}$$

from which we know that the following complex

$$\begin{aligned} R_d^{k-2}: H^*_{d^{h_{k-2}}}\rightarrow H^*_{d^{h_{k-2}}}, \end{aligned}$$

is also well defined on the corresponding harmonic space $\mathcal {H}^*_{d^{h_{k-2}}}$. Thus the spectral sequence $(d^{h_{k-2}},R_d^{k-2})$ degenerates at $E_3$ and

$$\begin{aligned} \mathcal {H}^*_{d^{h_{k-2}}+R_d^{k-2}}=\mathcal {H}^*_{d^{h_{k-2}}} \cap \ker \Box _{R_d^{k-2}}, \end{aligned}$$

which is equivalent to

$$\begin{aligned} \mathcal {H}^*_{d^{h_{k-3}}}=\ker \Box _{R_d^{k-2}+R_d^{k-1}}= \ker \Box _{R_d^{k-2}} \cap \Box _{R_d^{k-1}}. \end{aligned}$$

Repeating the above argument gives the proof of the real case, the complex case follows by a similar argument. $\square$

Remark

Not all nilpotent frames are admissible, we shall give a counterexample in the next section.

9 Further examples

9.1 A nilmanifold not satisfying Theorem 8.2

Let $\mathfrak {h}_7$ be the real 6-dimensional nilpotent Lie algebra with basis $\{e_1,\ldots , e_6\}$ such that

$$\begin{aligned} {[}e_1,e_2]=-\,e_4,\quad [e_1,e_3]=-\,e_5, \quad [e_2,e_3]=-\,e_6, \end{aligned}$$

the other brackets vanishing. Then, denoting by $\{e^1,\ldots , e^6\}$ the dual basis of $\{e_1,\ldots , e_6\}$, we obtain the following structure equations

$$\begin{aligned} \hbox {d}e^1=0,\quad \hbox {d}e^2=0, \quad \hbox {d}e^3=0,\quad \hbox {d}e^4=e^1\wedge e^2,\quad \hbox {d}e^5=e^1\wedge e^3,\quad \hbox {d}e^6=e^2\wedge e^3. \end{aligned}$$

Then, the simply connected Lie group $H_7$ whose Lie algebra is $\mathfrak {h}_7$ admits compact quotients $M=\Gamma \backslash H_7$. Define an almost complex structure $J_0$ on M by the following complex (1, 0)-coframe

$$\begin{aligned} \omega ^1=e^1+ie^2,\quad \omega ^2=-\,2(e^3+ie^4),\quad \omega ^3=-\,4(e^5+ie^6). \end{aligned}$$

Then

$$\begin{aligned} \hbox {d}\omega ^1=0,\quad d\omega ^2=\omega ^1\wedge \overline{\omega ^1},\quad \hbox {d}\omega ^3=\omega ^1\wedge \omega ^2+ \omega ^1\wedge \overline{\omega ^2}, \end{aligned}$$

that is $J_0$ is in fact integrable and by Ugarte [29] (see also [23]) any other complex structure on M is equivalent to $J_0$. Set

$$\begin{aligned} \omega ^1=\xi ^3,\quad \omega ^2=\xi ^2,\quad \omega ^3=\xi ^1. \end{aligned}$$

Then,

$$\begin{aligned} \hbox {d}\xi ^1=-\,\xi ^2\wedge \xi ^3+\xi ^3\wedge \overline{\xi ^2},\quad \hbox {d}\xi ^2=\xi ^3\wedge \overline{\xi ^3},\quad \hbox {d}\xi ^3=0 \end{aligned}$$

(9.1)

and $k=3$. The decomposition of $\wedge ^{1,0}T^*_M$ into vertical and horizontal subbundles reads as

$$\begin{aligned} \wedge ^{1,0}T^*_M=\langle \xi ^1\rangle \oplus \langle \xi ^2,\xi ^3\rangle . \end{aligned}$$

A direct computation taking into account (9.1) gives

$$\begin{aligned} R^1_{\overline{\partial }}= & {} -\,\overline{\xi ^2}\wedge \overline{\xi ^3} \wedge (\overline{\xi _1}\, \rfloor \,)+\xi ^3\wedge \overline{\xi ^2}\wedge (\xi _1\,\rfloor \,), R^2_{\overline{\partial }}=\xi ^3\wedge \overline{\xi ^3}\wedge (\xi _2\,\rfloor \,),\\ (R^1_{\overline{\partial }})^*= & {} -\,\overline{ \xi ^1}\wedge [(\overline{\xi _3}\,\rfloor \,) (\overline{\xi _2}\,\rfloor \,)]+\xi ^1\wedge [(\overline{\xi _2}\,\rfloor \,) ( \xi _3\,\rfloor \,)], (R^2_{\overline{\partial }})^*=\xi ^2\wedge [(\overline{\xi _3}\,\rfloor \,) (\xi _3\,\rfloor )], \end{aligned}$$

where $\{\xi _1,\xi _2,\xi _3\}$ satisfies $\xi ^r(\xi _s)=\delta ^r_s$.

Theorem 9.1

The Dolbeault cohomology group of $\Gamma \backslash H_7$ satisfies

$$\begin{aligned} \mathcal {H}^{1,1}_{\overline{\partial }} \simeq \mathrm{Span}_{\mathbb {C}}\,\langle \xi ^{1\bar{2}}, \xi ^{2\bar{3}}, \xi ^{2\bar{2}}+\xi ^{3\bar{1}}, \ \xi ^{2\bar{2}}+\xi ^{1\bar{3}}, \rangle . \end{aligned}$$

(9.2)

where $\xi ^{i\bar{j}}:=\xi ^i\wedge \overline{\xi ^j}$. In particular, $\mathcal {H}^{1,1}_{\overline{\partial }}$ does not satisfy Theorem 8.2.

Proof

Let

$$\begin{aligned} u=a_1\xi ^{1\bar{1}}+b_1\xi ^{2\bar{1}}+c_1\xi ^{3\bar{1}}+a_2\xi ^{1\bar{2}} +b_2\xi ^{2\bar{2}}+c_2\xi ^{3\bar{2}}+a_3\xi ^{1\bar{3}}+b_3\xi ^{2\bar{3}}+c_3\xi ^{3\bar{3}} \end{aligned}$$

be a harmonic (1, 1)-form. By our main theorem, one may assume that all the above coefficients are constants. Then, u is harmonic if and only if

$$\begin{aligned} \overline{\partial }u=\left( R^1_{\overline{\partial }}+R^2_{\overline{\partial }}\right) u= \overline{\partial }^* u= \left( (R^1_{\overline{\partial }})^*+(R^2_{\overline{\partial }})^*\right) u=0, \end{aligned}$$

which gives (9.2). Notice that $\mathcal {H}^{1,1}_{\overline{\partial }}$ satisfies Theorem 8.2 if and only if

$$\begin{aligned} \mathcal {H}^{1,1}_{\overline{\partial }} \subset \ker R^1_{\overline{\partial }} \cap \ker (R^1_{\overline{\partial }})^*\cap \ker R^2_{\overline{\partial }} \cap \ker (R^2_{\overline{\partial }})^* \end{aligned}$$

But obviously

$$\begin{aligned} R^2_{\overline{\partial }} (\xi ^{2\bar{1}}+\xi ^{3\bar{1}}) \ne 0, \end{aligned}$$

thus $\mathcal {H}^{1,1}_{\overline{\partial }}$ does not satisfy Theorem 8.2. $\square$

9.2 Nilpotent frame in a non-nilpotent Lie group

Let us look at the following example from [16].

Example

Consider the following group structure

$$\begin{aligned} (a, \lambda )*(t,z)=(a+t, \lambda +e^{2\pi i a} z), \end{aligned}$$

on $\mathbb {R}\times \mathbb {C}$. Then we know that

$$\begin{aligned} G:=(\mathbb {R}\times \mathbb {C}, *), \end{aligned}$$

is a Lie group. It is clear that

$$\begin{aligned} \Gamma :=\mathbb {Z}\times (\mathbb {Z}+i\mathbb {Z}) \end{aligned}$$

is a discrete subgroup of G such that the quotient space (with respect to the left action)

$$\begin{aligned} X:=\Gamma \backslash G, \end{aligned}$$

is a compact smooth manifold. It is clear that

$$\begin{aligned} \hbox {d}t, \ \ e^{-2\pi i t} \hbox {d}z, \end{aligned}$$

are G-invariant with respect to the left action of G. Put

$$\begin{aligned} e^1=r\hbox {d}t, \ e^2=\mathrm{Re} (e^{-2\pi i t} \hbox {d}z), \ e^3=\mathrm{Im} (e^{-2\pi i t} \hbox {d}z). \end{aligned}$$

we know that $\{e^1, e^2, e^3\}$ is a basis of the dual Lie algebra $\mathfrak {g}^*$ of G. Notice that if $z=x+iy$ then

$$\begin{aligned} e^2=(\cos 2\pi t) \, \hbox {d}x+(\sin 2\pi t) \, \hbox {d}y, \end{aligned}$$

and

$$\begin{aligned} e^3=-\,(\sin 2\pi t)\, \hbox {d}x+(\cos 2\pi t)\, \hbox {d}y. \end{aligned}$$

Thus

$$\begin{aligned} \hbox {d}e^1=0, \ \hbox {d}e^2=2\pi \,e^1 \wedge e^3, \ \hbox {d}e^3=-\,2\pi \, e^1 \wedge e^2, \end{aligned}$$

which gives

$$\begin{aligned} {[}\mathfrak {g}, [\mathfrak {g}, \mathfrak {g}]]=[\mathfrak {g}, \mathfrak {g}], \ \ [[\mathfrak {g}, \mathfrak {g}], [\mathfrak {g}, \mathfrak {g}]]=0. \end{aligned}$$

Thus $\mathfrak {g}$ is solvable but not nilpotent. But notice that

$$\begin{aligned} \sigma ^1:=\hbox {d}t, \ \ \sigma ^2:=\hbox {d}x, \ \ \sigma ^3:=\hbox {d}y, \end{aligned}$$

give a smooth frame of $T^*_X$ such that

$$\begin{aligned} \hbox {d}\sigma ^j =0, \ j=1,2,3. \end{aligned}$$

Thus

$$\begin{aligned} \Phi :=\{\sigma ^1, \sigma ^2, \sigma ^3\}, \end{aligned}$$

is a nilpotent frame of X and our main theorem implies that the de Rham cohomology of X is isomorphic to the exterior algebra generated by $\Phi$ (in fact it is easy to see that X is diffeomorphic to a real torus).