Global solvability and cohomology of tube structures on compact manifolds

Abstract

We introduce new techniques to study the differential complexes associated to tube structures on $M \times \mathbb {T}^m$ of corank m, in which M is a compact manifold and $\mathbb {T}^m$ is the m-torus. By systematically employing partial Fourier series, for complex tube structures, we completely characterize global solvability, in a given degree, in terms of a weak form of hypoellipticity, thus generalizing existing results and providing a broad answer to an open problem proposed by Hounie and Zugliani (Math Ann 369(3–4):1177–1209, 2017). We also obtain new results on the finiteness of the cohomology spaces in intermediate degrees. In the case of real tube structures, we extend an isomorphism for the cohomology spaces originally obtained by Dattori da Silva and Meziani (Math Nachr 289(17–18):2147–2158, 2016) in the case $M = \mathbb {T}^n$. Moreover, we establish necessary and sufficient conditions for the differential operator to have closed range in the first degree.

1 Introduction

We investigate the global solvability and the cohomology spaces of differential complexes associated with certain systems of first-order differential operators on compact manifolds. More precisely, let M be a smooth, compact, connected and oriented n-dimensional manifold, and let $\omega _1, \ldots , \omega _m$ be smooth, complex, closed 1-forms on M, and let $\mathbb {T}^m \doteq \mathbb {R}^m / 2 \pi \mathbb {Z}^m$. On $\Omega \doteq M \times \mathbb {T}^m$, we consider the involutive subbundle $\mathcal {V}\subset \mathbb {C}T \Omega$ whose sections are annihilated by

$$\begin{aligned} \zeta _k \doteq \textrm{d}x_k - \omega _k, \quad k \in \{1, \ldots , m\}, \end{aligned}$$

(1.1)

in which $x = (x_1, \ldots , x_m)$ denote the usual coordinates on $\mathbb {T}^m$. Such $\mathcal {V}$ gives rise to a complex of vector bundles and first-order differential operators over $\Omega$ [5, 10], here constructed as follows: for each $q \in \{0, \ldots , n\}$, let $\Lambda ^q$ denote the bundle of q-forms over M and $\underline{\Lambda }^q$ its pullback via the projection $\Omega \rightarrow M$. The smooth sections of the latter are locally written as

$$\begin{aligned} f = \mathop {{\mathop {\sum }\nolimits '}}\limits _{|J| = q} f_J(t,x) \textrm{d}t_J, \end{aligned}$$

(1.2)

in which $f_J \in \mathscr {C}^\infty (U \times \mathbb {T}^m)$ and $(U; t_1, \ldots , t_n)$ is some local chart of M. Denote by $\textrm{d}_t$ the exterior derivative in M and define a differential operator

$$\begin{aligned} \textrm{d}' \doteq \textrm{d}_t + \sum _{k = 1}^m \omega _k \wedge \frac{\partial }{\partial x_k}: \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q}) \longrightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q + 1}), \end{aligned}$$

(1.3)

that satisfies $\textrm{d}' \circ \textrm{d}' = 0$. Our goal is to investigate global solvability—that is, closedness of the range—of (1.3) in any degree $q \in \{0, \ldots , n - 1\}$, and to provide a better understanding of the smooth cohomology spaces

$$\begin{aligned} H^{q}_{\textrm{d}'}(\mathscr {C}^\infty (\Omega )) \doteq \frac{ \ker \{ \textrm{d}': \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q}) \longrightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q + 1}) \} }{ {{\,\mathrm{\textrm{ran}}\,}}\{ \textrm{d}': \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1}) \longrightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q}) \} }, \end{aligned}$$

(1.4)

for $q \in \{0, \ldots , n \}$ (with standard conventions regarding the endpoints¹).

Our motivation to work in this setting is an attempt to encompass the model in [3, 4, 8, 9] where the global solvability of $\textrm{d}'$ in $\Omega = M \times \mathbb {T}^1$ is studied and also the model in [6] where the authors were able to find isomorphisms for the cohomology spaces when $\Omega = \mathbb {T}^n \times \mathbb {T}^m$. Below, we summarize what we understand to be three of the main themes that keep intertwining throughout the paper, and their major consequences.

1.1 The relationship between solvability and regularity

Our first result is an equivalence between global solvability of $\textrm{d}'$ and a weak notion of hypoellipticity (Theorem 4.4). This is a generalization of [9, Corollary 7.2] that holds in arbitrary degree, for any smooth, complex tube structure of arbitrary corank, and thus provides a broad answer to the Open Problem 2 in [8]. All the steps of this implication are explained in Remark 4.3, Corollary 4.5 and Remark 4.6.

The main statement can be summarized as follows:

Theorem 1.1

For each $q \in \{0, \ldots , n - 1\}$, the following are equivalent:

• (1) $\textrm{d}': \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^q) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q + 1})$ has closed range.

• (2) For every $u \in \mathscr {D}'(\Omega ; \underline{\Lambda }^q)$ such that $\textrm{d}' u \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q + 1})$ there exists $v \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^q)$ such that $\textrm{d}' v = \textrm{d}' u$.

The second property above is what we will call almost global hypoellipticity $\mathrm {(AGH)}$.

1.2 Isomorphism formulas for the cohomology spaces

Concerning the cohomology spaces for real tube structures, we prove certain isomorphisms (5.10), similar to the main result in [6]. In that work, M is the n-torus, and their result roughly states that, under suitable conditions, we have

$$\begin{aligned} H^{q}_{\textrm{d}'}(\mathscr {C}^\infty (\Omega )) \cong \mathscr {C}^\infty (\mathbb {T}^r) \otimes H^q_{\textrm{dR}}(M), \end{aligned}$$

(1.5)

in which r is the rank of a group associated to $\omega _1, \ldots , \omega _m$. For a general M, however, our approach (see Sect. 5.1) generalizes and provides a better understanding of such result by decomposing the action of our operator in convenient subspaces. While the complete statement of our results requires a bit of notation, below we will provide rough versions of them for the convenience of the reader.

Assume that $\omega _1, \ldots , \omega _m$ are closed and real-valued 1-forms on M. We define the following subgroup² of $\mathbb {Z}^m$:

$$\begin{aligned} \Gamma _{\pmb {\omega }} \doteq \Big \{ \xi \in \mathbb {Z}^m \ ; \ \xi \cdot \pmb {\omega } \doteq \sum _{k = 1}^m \xi _k \omega _k \ \text { is an integral } 1\text {-form} \Big \}. \end{aligned}$$

We interpret the $\xi \in \mathbb {Z}^m$ as frequencies for the partial Fourier series with respect to $x \in \mathbb {T}^m$. Roughly speaking, by applying Fourier series to cohomology classes in $H^q_{\textrm{d}'}(\mathscr {C}^\infty (\Omega ))$ and splitting them into frequencies belonging to $\Gamma _{\pmb {\omega }}$ and to $\mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$, one is naturally led to the direct sum decomposition

$$\begin{aligned} H^{q}_{\textrm{d}'}(\mathscr {C}^\infty (\Omega )) \cong H^{q}_{\textrm{d}'}(\mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\Omega )) \oplus H^{q}_{\textrm{d}'}(\mathscr {C}^\infty _{\mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}}(\Omega )) \end{aligned}$$

(the definition of the summands can be found in Sect. 5.1). Then, as a result of a series of arguments in Sect. 5.1, we obtain the following representation of the first factor above (which, again, assume no hypotheses on the given tube structure):

$$\begin{aligned} H^{q}_{\textrm{d}'}(\mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\Omega )) \cong \mathscr {C}^\infty (\mathbb {T}^r) \otimes H^q_{\textrm{dR}}(M), \end{aligned}$$

where r is the rank of $\Gamma _{\pmb {\omega }}$ as a group. In order to reach (1.5), we are necessarily lead to the problem of determining conditions for the vanishing of the second factor $H^{q}_{\textrm{d}'}(\mathscr {C}^\infty _{\mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}}(\Omega ))$: this is done in Theorem 5.7, where, besides the obvious requirement of closedness of the range of $\textrm{d}'$, we find an unexpected obstruction (5.11) for the validity of (1.5). Therefore, the main result in [6] does not extend, in general, when M is not an n-torus.

Indeed, for surfaces, this obstruction—which is related both to the nature of the tube structure and the topology of M—is not present in any degree when M is either a 2-torus or a 2-sphere, but is present at $q = 1$ when M has genus ${\textsf{g}} \ge 2$. We show in Sect. 7.2 how to construct tube structures that satisfy all the hypotheses required by [6], but that do not satisfy (1.5). We conjecture that (5.11) holds in any degree when M is a compact Lie group and the tube structure is real, thus providing a complete generalization of the results in [6]. We plan to investigate this topic further in a forthcoming work.

1.3 Characterization of solvability in the first degree

Still assuming our 1-forms $\omega _1, \ldots \omega _m$ to be real, in Sect. 5.2 we provide a complete characterization of global solvability for the first degree of our complexes in terms of Diophantine conditions à la [3, 4]. Here, this is our main result, a restatement of Theorem 5.12:

Theorem 1.2

Assume $\omega _1, \ldots , \omega _m$ real-valued. The following are equivalent:

• (1) $\textrm{d}': \mathscr {C}^\infty (\Omega ) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^1)$ has closed range.

• (2) Given $\{ \beta _\nu \}_{\nu \in \mathbb {N}} \subset \mathscr {C}^\infty (M; \Lambda ^1)$ a sequence of closed integral 1-forms and $\{ \xi _\nu \}_{\nu \in \mathbb {N}} \subset \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$ such that $|\xi _\nu | \rightarrow \infty$, we have that $\{ | \xi _\nu |^\nu (\xi _\nu \cdot \pmb {\omega } - \beta _\nu ) \}_{\nu \in \mathbb {N}}$ is unbounded in $\mathscr {C}^\infty (M; \Lambda ^1)$.

The second condition above is called the property of weak non-simultaneous approximability for the collection $(\omega _1, \ldots , \omega _m)$, which is further discussed in Sect. 5.2.

The systematic use of the key Lemma 4.1 and its implications not only provides a better understanding of the different notions of global solvability present in previous works (that use the so-called compatibility conditions, see Remark 4.3) but also results into relatively short and straightforward proofs without the need of usual techniques, such as dualizing with the top degree, or the use of a priori inequalities. Indeed, our method automatically supply a formal solution (Definition 4.7), obtained by solving a system of simpler differential equations (with the help of compatibility conditions) that appears after performing the partial Fourier transform. The question then becomes when the formal solution is a true solution, and here the Diophantine condition plays a role.

1.4 Final results: finiteness theorems and results on surfaces

In Sects. 6 and 7, we revisit complex tube structures, and derive a handful of necessary conditions for finiteness of the cohomology spaces. The results we proved led us to conjecture whether $H^{q}_{\textrm{d}'}(\mathscr {C}^\infty (\Omega ))$ is finite dimensional only if it is isomorphic with $H^q_{\textrm{dR}}(M)$—true, notably, when (5.11) holds and $r = 0$ (Theorem 7.5), as well as in every other situation in which we were able to compute. For example, when M is a surface, we obtain a quite complete description (Sect. 7.1).

Convention. Except where explicitly stated, the 1-forms $\omega _1, \ldots , \omega _m$ are assumed to be complex-valued.

2 Preliminaries

2.1 Global solvability in abstract complexes and related concepts

Let X be a smooth, compact, connected and oriented manifold, and $\mathbb {E}$ be a complex vector bundle over X. The space $\mathscr {C}^\infty (X; \mathbb {E})$ of smooth sections of $\mathbb {E}$ carries its standard Fréchet topology; by endowing X with a Riemannian metric and $\mathbb {E}$ with a Hermitian metric, one may write it as the projective limit of a suitable sequence of Sobolev spaces of sections of $\mathbb {E}$. We also let $\mathscr {D}'(X; \mathbb {E})$ be the space of distribution sections of $\mathbb {E}$, which will be identified with the topological dual of $\mathscr {C}^\infty (X; \mathbb {E}^*)$.

Let $\mathbb {E}, \mathbb {F}$ be vector bundles over X and P a differential operator from $\mathbb {E}$ to $\mathbb {F}$.

Definition 2.1

We say that P is almost globally hypoelliptic (AGH) if

$$\begin{aligned} \forall u \in \mathscr {D}'(X; \mathbb {E}), \ Pu \in \mathscr {C}^\infty (X; \mathbb {F}) \Longrightarrow \exists v \in \mathscr {C}^\infty (X; \mathbb {E}) \text { such that } Pv = Pu. \end{aligned}$$

We have:

Theorem 2.2

If P is $\mathrm {(AGH)}$, then $P: \mathscr {C}^\infty (X; \mathbb {E}) \rightarrow \mathscr {C}^\infty (X; \mathbb {F})$ has closed range.

This result is proved in [2, Theorem 3.5] for scalar operators; its proof extends to vector-valued operators in a straightforward way, hence we omit it. A converse fails to hold even for very simple operators, but is valid for many classes of operators [1, 2], including (1.3), as we will prove in Theorem 4.4 below.

Let $\mathbb {G}$ be a third vector bundle over X and Q a differential operator from $\mathbb {F}$ to $\mathbb {G}$ such that $Q \circ P = 0$. We define two cohomology spaces

$$\begin{aligned} H_{P,Q}(\mathscr {F}(X)) \doteq \frac{\ker \{ Q: \mathscr {F}(X; \mathbb {F}) \longrightarrow \mathscr {F}(X; \mathbb {G}) \}}{{{\,\mathrm{\textrm{ran}}\,}}\{ P: \mathscr {F}(X; \mathbb {E}) \longrightarrow \mathscr {F}(X; \mathbb {F}) \}}, \quad \mathscr {F} = \hbox {} \mathscr {C}^\infty ~\hbox {or}~\mathscr {D}'. \end{aligned}$$

Our goal is to understand these two cohomology spaces separately, as well as their relationship. The inclusions $\mathscr {C}^\infty (X; *) \hookrightarrow \mathscr {D}'(X; *)$ ($*= \mathbb {E}, \mathbb {F}, \mathbb {G}$) induce a linear morphism

$$\begin{aligned} H_{P,Q}(\mathscr {C}^\infty (X)) \longrightarrow H_{P,Q}(\mathscr {D}'(X)) \end{aligned}$$

(2.1)

which may be neither injective nor surjective. However,

$$\begin{aligned} \hbox {} P~\hbox {is~} \text {(AGH)} \Longleftrightarrow (2.1) \text { is injective}, \end{aligned}$$

being therefore independent of Q, and a property of P alone. One should recall the traditional notion of global hypoellipticity for complexes, that is:

$$\begin{aligned}&\forall f \in \mathscr {D}'(X; \mathbb {F}), \ Qf \in \mathscr {C}^\infty (X; \mathbb {G}) \Longrightarrow \nonumber \\&{\exists u \in \mathscr {D}'(X; \mathbb {E})} \hbox { such that } f - Pu \in \mathscr {C}^\infty (X; \mathbb {F}). \end{aligned}$$

(2.2)

Since $Q (f - Pu) = Qf$, (2.2) implies that Q is $\mathrm {(AGH)}$. Additionally, we have that

$$\begin{aligned} \forall f \in \mathscr {D}'(X; \mathbb {F}), \ Qf = 0 \Longrightarrow \exists u \in \mathscr {D}'(X; \mathbb {E}) \text { such that } f - Pu \in \mathscr {C}^\infty (X; \mathbb {F}), \end{aligned}$$

which is equivalent to the surjectivity of (2.1). We state this result more precisely:

Proposition 2.3

Property (2.2) holds if and only if Q is $\mathrm {(AGH)}$ and (2.1) is onto.

The transpose is a differential operator from $\mathbb {F}^*$ to $\mathbb {E}^*$, yielding new maps

the latter being the restriction of the former. In the presence of a second operator Q satisfying $Q \circ P = 0$, we want to determine necessary conditions on $f \in \mathscr {C}^\infty (X; \mathbb {F})$ so as to be able to solve $Pu = f$ with $u \in \mathscr {C}^\infty (X; \mathbb {E})$. Obviously, we must have

$$\begin{aligned} f \in \ker \{ Q: \mathscr {C}^\infty (X; \mathbb {F}) \longrightarrow \mathscr {C}^\infty (X; \mathbb {G}) \}. \end{aligned}$$

(2.3)

Moreover, if $v \in \mathscr {D}'(X; \mathbb {F}^*)$ is such that , then

that is,

(2.4)

However, by Functional Analysis, the annihilator in (2.4) equals the closure of ${{\,\mathrm{\textrm{ran}}\,}}\{ P: \mathscr {C}^\infty (X; \mathbb {E}) \rightarrow \mathscr {C}^\infty (X; \mathbb {F}) \}$, and since $Q \circ P = 0$, the range of P is contained in $\ker \{ Q: \mathscr {C}^\infty (X; \mathbb {F}) \rightarrow \mathscr {C}^\infty (X; \mathbb {G}) \}$, which is closed in $\mathscr {C}^\infty (X; \mathbb {F})$. We conclude:

hence the compatibility condition (2.3) is redundant in light of (2.4). We therefore introduce the following definition in spite of the presence of the operator Q.

Definition 2.4

We say that P is globally solvable if for every $f \in \mathscr {C}^\infty (X; \mathbb {F})$ satisfying

there exists $u \in \mathscr {C}^\infty (X; \mathbb {E})$ such that $Pu = f$.

In other words, P is globally solvable if and only if the range of P is closed.

2.2 Partial Fourier series for sections of $\underline{\Lambda }^{q}$

Given $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$, for each $\xi \in \mathbb {Z}^m$, we define an element ${\hat{f}}_\xi \in \mathscr {C}^\infty (M; \Lambda ^q)$ as follows: if $U \subset M$ is a coordinate open set in which f is written as (1.2) then

$$\begin{aligned} {\hat{f}}_\xi (t) \doteq \mathop {{\mathop {\sum }\nolimits '}}\limits _{|J| = q} {\hat{f}}_J(t,\xi ) \textrm{d}t_J = \mathop {{\mathop {\sum }\nolimits '}}\limits _{|J| = q} \left( \int _{\mathbb {T}^m} e^{-i x \xi } f_J(t, x) \textrm{d}x \right) \textrm{d}t_J, \quad t \in U. \end{aligned}$$

Note that since $\underline{\Lambda }_{(t,x)}= \Lambda _t$ for every $x \in \mathbb {T}^m$, each $\textrm{d}t_J$ can be thought as a section of either $\Lambda ^q$ or $\underline{\Lambda }^q$. One can check that the construction above is independent of the choice of coordinates on U, hence defines well a differential form of degree q in M. It is useful to regard ${\hat{f}}_\xi$ as a current on M, that is, we define ${\hat{f}}_\xi : \mathscr {C}^\infty (M; \Lambda ^{n - q}) \rightarrow \mathbb {C}$ by

$$\begin{aligned} g \mapsto \langle {\hat{f}}_\xi , g \rangle _M \doteq \langle f, e^{-ix \xi } \wedge g \wedge \textrm{d}x \rangle = \int _\Omega f \wedge e^{-ix \xi } \wedge g \wedge \textrm{d}x, \end{aligned}$$

(2.5)

with $\textrm{d}x \doteq \textrm{d}x_1 \wedge \cdots \wedge \textrm{d}x_m$; both definitions yield the same object. This extends our construction to q-currents, yielding linear maps

$$\begin{aligned} \mathcal {F}_\xi : \left\{ \begin{array}{c c c} \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q}) &{} \longrightarrow &{} \mathscr {C}^\infty (M; \Lambda ^q) \\ \mathscr {D}'(\Omega ; \underline{\Lambda }^{q}) &{} \longrightarrow &{} \mathscr {D}'(M; \Lambda ^q) \end{array} \right. \end{aligned}$$

defined by the assignment $f \mapsto {\hat{f}}_\xi$.

Proposition 2.5

The map $\mathcal {F}_\xi : \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q}) \rightarrow \mathscr {C}^\infty (M; \Lambda ^q)$ is continuous.

Going in the other direction we define linear maps

$$\begin{aligned} \mathcal {E}_\xi : \left\{ \begin{array}{c c c} \mathscr {C}^\infty (M; \Lambda ^q) &{} \longrightarrow &{} \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q}) \\ \mathscr {D}'(M; \Lambda ^q) &{} \longrightarrow &{} \mathscr {D}'(\Omega ; \underline{\Lambda }^{q}) \end{array} \right. \end{aligned}$$

(2.6)

by $f \mapsto (2 \pi )^{-m} e^{ix \xi } \wedge f$.

Lemma 2.6

For every $\xi \in \mathbb {Z}^m$, we have that $\mathcal {F}_\xi \circ \mathcal {E}_\xi$ is the identity on $\mathscr {C}^\infty (M; \Lambda ^q)$ and $\mathcal {F}_\xi \circ \mathcal {E}_\eta = 0$ if $\eta \ne \xi$, for every $q \in \{0,\ldots , n\}$.

Proposition 2.7

Given $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ we have

$$\begin{aligned} f = \frac{1}{(2 \pi )^m} \sum _{\xi \in \mathbb {Z}^m} e^{ix \xi } {\hat{f}}_\xi \end{aligned}$$

(2.7)

with convergence in $\mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$. In particular, $f = 0$ if and only if ${\hat{f}}_\xi = 0$ for all $\xi \in \mathbb {Z}^m$.

2.3 Fourier analysis of $\textrm{d}'$

Let $\pmb {\omega } \doteq (\omega _1, \ldots , \omega _m)$ with each $\omega _k$ being a complex, closed 1-form on M. Given $\xi \in \mathbb {Z}^m$, we write

$$\begin{aligned} \xi \cdot \pmb {\omega } \doteq \sum _{k = 1}^m \xi _k \omega _k \in \mathscr {C}^\infty (M; \Lambda ^1) \end{aligned}$$

and define $\textrm{d}'_\xi : \mathscr {C}^\infty (M; \Lambda ^q) \rightarrow \mathscr {C}^\infty (M; \Lambda ^{q + 1})$ by

$$\begin{aligned} \textrm{d}'_\xi f \doteq \textrm{d}f + i (\xi \cdot \pmb {\omega }) \wedge f, \end{aligned}$$

a first-order differential operator that satisfies $\textrm{d}'_\xi \circ \textrm{d}'_\xi = 0$, thus forming a complex whose smooth cohomology spaces we denote by

$$\begin{aligned} H_\xi ^q (\mathscr {C}^\infty (M)) \doteq \frac{\ker \{ \textrm{d}'_\xi : \mathscr {C}^\infty (M; \Lambda ^q) \longrightarrow \mathscr {C}^\infty (M; \Lambda ^{q + 1}) \}}{{{\,\mathrm{\textrm{ran}}\,}}\{ \textrm{d}'_\xi : \mathscr {C}^\infty (M; \Lambda ^{q - 1}) \longrightarrow \mathscr {C}^\infty (M; \Lambda ^q) \}}, \quad q \in \{1, \ldots , n\}\nonumber \\ \end{aligned}$$

(2.8)

and, as usual, $H_\xi ^0(\mathscr {C}^\infty (M)) \doteq \ker \{ \textrm{d}'_\xi : \mathscr {C}^\infty (M) \longrightarrow \mathscr {C}^\infty (M; \Lambda ^{ 1}) \}$.

In the next section, we start a deeper study of these zero-order perturbations of the de Rham complex. For the moment, we focus on their formal aspects related to the Fourier analysis of the complex $\textrm{d}'$. The proofs of the next results are standard.

Lemma 2.8

The following transposition formulas hold.

• (1) For $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ and $g \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{n - q - 1})$, we have

  $$\begin{aligned} \int _\Omega \textrm{d}' f \wedge g \wedge \textrm{d}x = (-1)^{q + 1} \int _\Omega f \wedge \textrm{d}' g \wedge \textrm{d}x. \end{aligned}$$

• (2) For $f \in \mathscr {C}^\infty (M; \Lambda ^q)$ and $g \in \mathscr {C}^\infty (M; \Lambda ^{n - q - 1})$, we have

  $$\begin{aligned} \int _M \textrm{d}'_\xi f \wedge g = (-1)^{q + 1} \int _M f \wedge \textrm{d}'_{-\xi } g \end{aligned}$$

  for every $\xi \in \mathbb {Z}^m$.

Corollary 2.9

For each $\xi \in \mathbb {Z}^m$, the following commutation relations hold:

• (1) $\textrm{d}' \circ \mathcal {E}_\xi = \mathcal {E}_\xi \circ \textrm{d}'_\xi$;

• (2) $\mathcal {F}_\xi \circ \textrm{d}' = \textrm{d}'_\xi \circ \mathcal {F}_\xi$.

3 Zero-order perturbations of the de Rham complex

Motivated by the introduction of the operators $\textrm{d}'_\xi$ in Sect. 2.3, we discuss a general class of operators on M that are perturbations of the exterior derivative. Let $\omega$ be a complex, smooth, closed 1-form on M, and define, for $q \in \{0, \ldots , n\}$:

$$\begin{aligned} \textrm{D}_\omega \doteq \textrm{d}+ i \omega \wedge \cdot : \mathscr {C}^\infty (M; \Lambda ^q) \longrightarrow \mathscr {C}^\infty (M; \Lambda ^{q + 1}). \end{aligned}$$

(3.1)

Note that $\textrm{d}'_\xi = \textrm{D}_{\omega }$ when $\omega = \xi \cdot \pmb {\omega }$. These are first-order differential operators, satisfying $\textrm{D}_\omega \circ \textrm{D}_\omega = 0$ since $\omega$ is closed. The operators $\textrm{D}_\omega$ define a differential complex, which is elliptic since $\textrm{D}_\omega$ has the same principal symbol as $\textrm{d}$. Such a differential complex, however, does not come from an involutive structure on M (for instance, $\textrm{D}_\omega (1) = i\omega \ne 0$), hence we cannot apply solvability results from this theory to it. Given any open set $U \subset M$, we define

$$\begin{aligned} H_\omega ^q(\mathscr {F}(U)) \doteq \frac{\ker \{ \textrm{D}_\omega : \mathscr {F}(U; \Lambda ^q) \longrightarrow \mathscr {F}(U; \Lambda ^{q + 1}) \}}{{{\,\mathrm{\textrm{ran}}\,}}\{ \textrm{D}_\omega : \mathscr {F}(U; \Lambda ^{q - 1}) \longrightarrow \mathscr {F}(U; \Lambda ^q) \}}, \quad \mathscr {F} = {\mathscr {C}^\infty \text { or } \mathscr {D}'}.\qquad \end{aligned}$$

(3.2)

In particular, $H_\xi ^q(\mathscr {C}^\infty (M)) = H_{\omega }^q(\mathscr {C}^\infty (M))$ when $\omega =\xi \cdot \pmb {\omega }$. If $\omega$ is exact in U, say $\textrm{d}\phi = \omega |_U$ for a $\phi \in \mathscr {C}^\infty (U)$, then $\textrm{d}e^{i \phi } = i e^{i \phi } \omega |_U$ and

$$\begin{aligned} \textrm{d}(e^{i \phi } f) = \textrm{d}e^{i \phi } \wedge f + e^{i \phi } \textrm{d}f = e^{i \phi } \textrm{D}_\omega f \end{aligned}$$

whatever $f \in \mathscr {D}'(U; \Lambda ^q)$. In particular, for $q = 0$, if U is connected,

$$\begin{aligned} f \in \mathscr {D}'(U), \ \textrm{D}_\omega f = 0 \Longrightarrow f = \mathrm {const.} \cdot e^{-i \phi }, \end{aligned}$$

(3.3)

thus the sheaves of homogeneous solutions of $\textrm{D}_\omega$ in smooth functions and in distributions are one and the same; we will denote this sheaf by $\mathscr {S}_\omega$. A crucial feature is that $\mathscr {S}_\omega$ is locally isomorphic with the constant sheaf $\mathscr {S}_0$—its stalks are copies of $\mathbb {C}$—but not globally in general (see below). We have commutative diagrams

and the same holds for smooth sections. Since $e^{i \phi } \ne 0$ on U, we conclude that

$$\begin{aligned} H_\omega ^q(\mathscr {F}(U)) \cong H_{\textrm{dR}}^q(U), \quad \mathscr {F} = \hbox {} \mathscr {C}^\infty ~\hbox {or}~\mathscr {D}', \end{aligned}$$

(3.4)

the right-hand side standing for the usual de Rham cohomology space.

Let $\mathscr {F}_q$ denote the sheaf of germs of smooth or distributional sections (depending on the case) of $\Lambda ^q$, that is, forms and currents. We have the exactness in degree $q \in \{1, \ldots , n\}$ of the sequence

Indeed, by the Poincaré Lemma, every point $t \in M$ has a neighborhood in which $\omega$ is exact, thus we can apply the isomorphism (3.4) and use that the exterior derivative $\textrm{d}$ is locally solvable at t in degree $q \in \{1, \ldots , n\}$ (this is usually stated for smooth forms, but also holds for currents). In particular, in both cases the sequence (3.5) provides fine resolution of $\mathscr {S}_\omega$, so on any open set U, we have

$$\begin{aligned} H_{\omega }^q(\mathscr {C}^\infty (U)) \cong H_{\omega }^q(\mathscr {D}'(U)) \cong H^q(U; \mathscr {S}_\omega ), \end{aligned}$$

the right-hand side denoting the q-th cohomology space with values in the sheaf $\mathscr {S}_\omega$. Specifically, the natural homomorphism of sheaves $\Phi ^0: \mathscr {C}^\infty _0 \rightarrow \mathscr {D}'_0$ induces homomorphisms $\Phi ^q: \mathscr {C}^\infty _q \rightarrow \mathscr {D}'_q$, forming a homomorphism between resolutions

Hence, for each $q \in \{0, \ldots , n\}$, the map

$$\begin{aligned} \Phi ^q_*: H_{\omega }^q(\mathscr {C}^\infty (U)) \longrightarrow H_{\omega }^q(\mathscr {D}'(U)), \quad U \subset M, \end{aligned}$$

(3.6)

which is precisely the one induced by the inclusion map $\mathscr {C}^\infty (U; \Lambda ^q) \hookrightarrow \mathscr {D}'(U; \Lambda ^q)$ on the quotients, is an isomorphism³—we have just proved a special case of the so-called Atiyah–Bott Lemma. Unwinding quotients in (3.2), one deduces that:

Theorem 3.1

Given an open set $U \subset M$, we have that:

• (1) For every $f \in \mathscr {C}^\infty (U; \Lambda ^q)$ such that there exists $u \in \mathscr {D}'(U; \Lambda ^{q - 1})$ with $\textrm{D}_\omega u = f$, there exists $v \in \mathscr {C}^\infty (U; \Lambda ^{q - 1})$ such that $\textrm{D}_\omega v = f$. I.e., $\textrm{D}_\omega |_U$ is $\mathrm {(AGH)}$.

• (2) For every $f \in \mathscr {D}'(U; \Lambda ^q)$ satisfying $\textrm{D}_\omega f = 0$, there exist $g \in \mathscr {C}^\infty (U; \Lambda ^q)$ and $u \in \mathscr {D}'(U; \Lambda ^{q - 1})$ such that $f - g = \textrm{D}_\omega u$.

Proof

Clearly, the first claim is equivalent to the injectivity of (3.6), while the second one is equivalent to its surjectivity—both of them established above. $\square$

In particular, $\textrm{D}_\omega$ is $\mathrm {(AGH)}$ in each degree. On time, we recall that, by elliptic theory, all the cohomology spaces $H_{\omega }^q(\mathscr {C}^\infty (M))$ are finite dimensional since M is compact. Both properties will be used heavily, often without mention, from here on.

Another consequence of (3.3) is that, given a coordinate ball $U \subset M$, a function in $\mathscr {S}_\omega (U)$ either vanishes identically or is never zero. In particular, the zero set of a global homogeneous solution $f \in \mathscr {S}_\omega (M)$ is both open and closed, so by connectedness, such an f also either vanishes everywhere or not at all.

Lemma 3.2

Either $\mathscr {S}_\omega (M) = \{0\}$ or $\dim _\mathbb {C}\mathscr {S}_\omega (M) = 1$. The latter case happens if and only if $\mathscr {S}_\omega$ is isomorphic with the constant sheaf $\mathscr {S}_0$.

Proof

Suppose there exists $f \in \mathscr {S}_\omega (M)$ non-zero. By the previous remarks, f never vanishes. We will prove that any $g \in \mathscr {S}_\omega (M)$ is a constant times f. On an open set $U \subset M$ in which $\omega$ is exact, we have by (3.3) that $f|_U = c_1 e^{-i \phi }$ and $g|_U = c_2 e^{-i \phi }$ for some constants $c_1, c_2 \in \mathbb {C}$ with $c_1 \ne 0$. Hence $(g / f)|_U = c_1^{-1} c_2$, and, since such open sets cover M, we reach the conclusion that g/f is locally constant: by connectedness, this must be actually a constant function. Now, we prove that multiplication by f defines a sheaf isomorphism $\mathscr {S}_0 \rightarrow \mathscr {S}_\omega$, a claim that we check locally. Given $t \in M$, take a coordinate open ball $U \subset M$ around it. We have

$$\begin{aligned} h \in \mathscr {S}_0(U) \Longrightarrow h = \mathrm {const.} \Longrightarrow h \cdot f|_U = \mathrm {const.} \cdot f|_U \in \mathscr {S}_\omega (U), \end{aligned}$$

so we have a monomorphism $\mathscr {S}_0(U) \rightarrow \mathscr {S}_\omega (U)$. But, since $\omega |_U$ has a primitive, any $g \in \mathscr {S}_\omega (U)$ is a multiple of $f|_U$, hence that monomorphism is also surjective. $\square$

Corollary 3.3

If $\omega$ is exact then $\mathscr {S}_\omega \cong \mathscr {S}_0$.

Proof

If $\phi \in \mathscr {C}^\infty (M)$ is a primitive of $\omega$, then $\textrm{D}_{\omega } (e^{-i \phi }) = 0$, i.e., $e^{-i \phi } \in \mathscr {S}_\omega (M)$. $\square$

Remark 3.4

The sheaf $\mathscr {S}_\omega$ is always locally isomorphic to $\mathscr {S}_0$; the isomorphism can be made global precisely when $\mathscr {S}_\omega$ admits a non-vanishing global section.

Lemma 3.5

Given two closed 1-forms $\omega _1, \omega _2$ on M such that $\mathscr {S}_{\omega _1}(M) \ne \{0\} \ne \mathscr {S}_{\omega _2}(M)$, we have that $\mathscr {S}_{\omega _1 + \omega _2}(M) \ne 0$.

Proof

If $f_1$ (resp. $f_2$) is a section of $\mathscr {S}_{\omega _1}$ (resp. $\mathscr {S}_{\omega _2}$) then $f_1 f_2$ is a section of $\mathscr {S}_{\omega _1 + \omega _2}$. Indeed:

$$\begin{aligned} \textrm{d}(f_1 f_2)= & {} \textrm{d}f_1 \wedge f_2 + f_1 \wedge \textrm{d}f_2 = -i \omega _1 \wedge f_1 \wedge f_2 - i f_1 \wedge \omega _2 \wedge f_2\\= & {} - i (\omega _1 + \omega _2) \wedge (f_1 f_2). \end{aligned}$$

In particular, if $f_1 \in \mathscr {S}_{\omega _1}(M)$ and $f_2 \in \mathscr {S}_{\omega _2}(M)$ are both non-zero then $f_1 f_2 \in \mathscr {S}_{\omega _1 + \omega _2}(M)$ is non-zero. $\square$

Theorem 3.6

The set

$$\begin{aligned} \mathcal {Z}_M \doteq \{ [\omega ] \in H^1_{\textrm{dR}}(M) \ ; \ \mathscr {S}_\omega \cong \mathscr {S}_0 \} = \{ [\omega ] \in H^1_{\textrm{dR}}(M) \ ; \ \mathscr {S}_\omega (M) \ne \{0\} \} \end{aligned}$$

is a group.

Proof

Notice that $\mathcal {Z}_M$ is well-defined: if $\omega , \omega ^\bullet$ are two closed 1-forms in the same cohomology class such that $\mathscr {S}_\omega \cong \mathscr {S}_0$ then also $\mathscr {S}_{\omega ^\bullet } \cong \mathscr {S}_0$. Indeed, in that case there exists $\phi \in \mathscr {C}^\infty (M)$ such that $\omega ^\bullet = \omega + \textrm{d}\phi$: since $\mathscr {S}_\omega \cong \mathscr {S}_0$ by assumption and $\mathscr {S}_{\textrm{d}\phi } \cong \mathscr {S}_0$ by Corollary 3.3, it follows from Lemmas 3.5 and 3.2 that $\mathscr {S}_{\omega ^\bullet } \cong \mathscr {S}_0$.

The same argument shows that $\mathcal {Z}_M$ is an additive submonoid of $H^1_{\textrm{dR}}(M)$. Regarding inverses, if $[\omega ] \in \mathcal {Z}_M$ then there exists a non-vanishing $f \in \mathscr {S}_\omega (M)$; clearly,

$$\begin{aligned} \textrm{d}f + i f \omega = 0 \Longrightarrow \textrm{d}(1/f) - i (1/f) \omega = 0. \end{aligned}$$

Therefore, 1/f is a non-vanishing element of $\mathscr {S}_{-\omega }(M)$, so $-[\omega ] \in \mathcal {Z}_M$. $\square$

Remark 3.7

Recall that a real closed 1-form $\alpha$ is integral if $\int _\gamma \alpha \in 2 \pi \mathbb {Z}$ for every 1-cycle $\gamma$ in M. It follows from [4, Lemma 2.1] that

$$\begin{aligned} \mathcal {Z}_M = \{ [\omega ] \in H^1_{\textrm{dR}}(M) \ ; \ \textsf{Re}\omega \text { is integral and } \textsf{Im}\omega \text { is exact} \}. \end{aligned}$$

(3.7)

4 Formal characterization of the closure of ${{\,\mathrm{\textrm{ran}}\,}}\textrm{d}'$

In this section, we address the issue of formal solvability (i.e., at the level of partial Fourier series) and relate it with the notion of global solvability in degree $q\in \{1, \ldots , n\}$. Now that we have all the tools at our disposal, the proofs are pretty straightforward.

Lemma 4.1

For $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$, the following are equivalent:

• (1) f belongs to the closure of ${{\,\mathrm{\textrm{ran}}\,}}\{\textrm{d}': \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1}) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q}) \}$;

• (2) ${\hat{f}}_\xi$ is $\textrm{d}'_\xi$-exact for every $\xi \in \mathbb {Z}^m$.

Remark 4.2

The notion of ${\hat{f}}_\xi$ being $\textrm{d}'_\xi$-exact is unambiguous: if $\textrm{d}'_\xi u_\xi = {\hat{f}}_\xi$ is solvable in distributions, we can always find a smooth solution since $\textrm{d}'_\xi$ is $\mathrm {(AGH)}$ (Theorem 3.1).

Proof

Let $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ and suppose there exists $\{ u_\nu \}_{\nu \in \mathbb {N}} \subset \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1})$ such that $\textrm{d}' u_\nu \rightarrow f$ in $\mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$. By Proposition 2.5 we have that

$$\begin{aligned} \textrm{d}'_\xi \mathcal {F_\xi } (u_\nu ) = \mathcal {F_\xi } (\textrm{d}' u_\nu ) \longrightarrow {\hat{f}}_\xi \quad \hbox { in}\ \mathscr {C}^\infty (M; \Lambda ^q). \end{aligned}$$

Therefore, ${\hat{f}}_\xi$ belongs to the closure of ${{\,\mathrm{\textrm{ran}}\,}}\{ \textrm{d}'_\xi : \mathscr {C}^\infty (M; \Lambda ^{q - 1}) \rightarrow \mathscr {C}^\infty (M; \Lambda ^q) \}$, which is already closed in $\mathscr {C}^\infty (M; \Lambda ^q)$—for instance, by Theorems 2.2 and 3.1—hence:

$$\begin{aligned} {\hat{f}}_\xi \in {{\,\mathrm{\textrm{ran}}\,}}\{\textrm{d}'_\xi : \mathscr {C}^\infty (M; \Lambda ^{q - 1}) \longrightarrow \mathscr {C}^\infty (M; \Lambda ^q) \}, \quad \forall \xi \in \mathbb {Z}^m. \end{aligned}$$

(4.1)

Conversely, if $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ is such that (4.1) holds, then for each $\xi \in \mathbb {Z}^m$ there exists $u_\xi \in \mathscr {C}^\infty (M; \Lambda ^{q - 1})$ such that $\textrm{d}'_\xi u_\xi = {\hat{f}}_\xi$, hence in the topology of $\mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ we have

$$\begin{aligned} f = \frac{1}{(2\pi )^m} \sum _{\xi \in \mathbb {Z}^m} e^{ix \xi } \wedge \textrm{d}'_\xi u_\xi = \lim _{\nu \rightarrow \infty } \textrm{d}' \Big ( \frac{1}{(2\pi )^m} \sum _{|\xi | \le \nu } e^{ix \xi } \wedge u_\xi \Big ), \end{aligned}$$

which proves that f lies in the closure of ${{\,\mathrm{\textrm{ran}}\,}}\{ \textrm{d}': \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1}) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q}) \}$. $\square$

Remark 4.3

Lemma 4.1 helps to settle the question of equivalence between different formulations of compatibility conditions appearing in the literature (hence, the equivalence between different notions of (smooth) global solvability).

Consider, for instance, the space $E \subset \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^1)$ determined by the compatibility conditions in [8, 9] (which deal with the case $m = 1$, $q = 1$). Given $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^1)$, it follows easily from de Rham Theorem that, if ${\hat{f}}_\xi$ is $\textrm{d}'_\xi$-exact for every $\xi \in \mathbb {Z}^m$, then $f \in E$; whereas the converse follows from results in [8, 9]: if $f \in E$, a solution to $\textrm{d}'_\xi u_\xi = {\hat{f}}_\xi$ is obtained for every $\xi \in \mathbb {Z}^m$, first in some covering space of M, and then in M by a convenient choice of initial conditions. In particular, E equals the closure of ${{\,\mathrm{\textrm{ran}}\,}}\{ \textrm{d}': \mathscr {C}^\infty (\Omega ) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{1}) \}$ in that case.

The previous lemma yields our first major result; which, together with Theorem 2.2, entails Theorem 1.1.

Theorem 4.4

If (1.3) has closed range, then it is $\mathrm {(AGH)}$.

Proof

Suppose that $\textrm{d}': \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q}) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q + 1})$ has closed range and let $u \in \mathscr {D}'(\Omega ; \underline{\Lambda }^{q})$ be such that $f \doteq \textrm{d}' u \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q + 1})$. Then ${\hat{f}}_\xi = \textrm{d}'_\xi {\hat{u}}_\xi$, that is, ${\hat{f}}_\xi$ is $\textrm{d}'_\xi$-exact for every $\xi \in \mathbb {Z}^m$. By Lemma 4.1, f belongs to ${{\,\mathrm{\textrm{ran}}\,}}\{ \textrm{d}': \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q}) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q + 1}) \}$, so there exists $v \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ such that $\textrm{d}' v = f = \textrm{d}' u$. $\square$

It generalizes [9, Corollary 7.2] (and preceding results):

Corollary 4.5

If the operator $\textrm{d}': \mathscr {C}^\infty (\Omega ) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^1)$ satisfies the property

$$\begin{aligned} \forall u \in \mathscr {D}'(\Omega ), \ \textrm{d}' u = 0 \Longrightarrow u = \mathrm {const.}, \end{aligned}$$

(4.2)

then it has closed range if and only if it is globally hypoelliptic.

Proof

This is an immediate consequence of Theorems 2.2 and 4.4, since global hypoellipticity is equivalent to $\mathrm {(AGH)}$ when (4.2) holds. $\square$

Remark 4.6

Notice that (4.2) holds in the case considered in [9, Corollary 7.2] ($m = 1$, $\omega = ib$ with b real, closed and non-exact) thanks to [4, Lemma 2.2]. See also further discussion below about the group $\Gamma _{\pmb {\omega }}$.

Definition 4.7

We say that an $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ is formally solvable if for each $\xi \in \mathbb {Z}^m$ there exists $u_\xi \in \mathscr {C}^\infty (M; \Lambda ^{q - 1})$ such that $\textrm{d}'_\xi u_\xi = {\hat{f}}_\xi$.

A sufficient condition for that to happen is that there exists u in $\mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1})$—or even in $\mathscr {D}'(\Omega ; \underline{\Lambda }^{q - 1})$—such that $\textrm{d}' u = f$: Lemma 4.1 entails the following converse.

Corollary 4.8

The following are equivalent:

• (1) $\textrm{d}': \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1}) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ has closed range.

• (2) For every formally solvable $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ there exists $u \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1})$ such that $\textrm{d}' u = f$.

Before we move on, we must introduce an important set of frequencies. For $\pmb {\omega } = (\omega _1, \ldots , \omega _m)$, the mapping $\xi \in \mathbb {Z}^m \mapsto [\xi \cdot \pmb {\omega }] \in H^1_{\textrm{dR}}(M)$ is a homomorphism of groups, hence

$$\begin{aligned} \Gamma _{\pmb {\omega }} \doteq \{ \xi \in \mathbb {Z}^m \ ; \ [\xi \cdot \pmb {\omega }] \in \mathcal {Z}_M \} \end{aligned}$$

(4.3)

is a subgroup of $\mathbb {Z}^m$. In particular, either one of the three mutually exclusive alternatives hold:

• (1) $\Gamma _{\pmb {\omega }} = \{0 \}$;

• (2) $\Gamma _{\pmb {\omega }}$ is infinite and proper; or

• (3) $\Gamma _{\pmb {\omega }} = \mathbb {Z}^m$.

Using (3.7) one has, for instance, that:

$$\begin{aligned} \Gamma _{\pmb {\omega }} = \{0 \}&\Longleftrightarrow \hbox {} \textsf{Re}(\xi \cdot \pmb {\omega })~\hbox {non-integral or}~\textsf{Im}(\xi \cdot \pmb {\omega })~\hbox {non-exact}, \forall \xi \in \mathbb {Z}^m {\setminus } \{0\} \\&\Longrightarrow \textsf{Re}\omega _k \text { non-integral or } \textsf{Im}\omega _k~ \hbox {non-exact}, \forall k \in \{1, \ldots , m\}; \end{aligned}$$

while

$$\begin{aligned} \Gamma _{\pmb {\omega }} = \mathbb {Z}^m&\Longleftrightarrow \hbox {} \textsf{Re}(\xi \cdot \pmb {\omega })~\hbox {integral and}~\textsf{Im}(\xi \cdot \pmb {\omega })~\hbox {exact}, \forall \xi \in \mathbb {Z}^m \\&\Longleftrightarrow {\textsf{Re}\omega _k~\hbox {integral and}~\textsf{Im}\omega _k~\hbox {exact}, \forall k \in \{1, \ldots , m\}.} \end{aligned}$$

5 Applications to real structures

Throughout this section, $\omega _1, \ldots , \omega _m$ are assumed real. In this case,

$$\begin{aligned} \Gamma _{\pmb {\omega }} = \{ \xi \in \mathbb {Z}^m \ ; \ \xi \cdot \pmb {\omega } \text { is integral} \}. \end{aligned}$$

5.1 Isomorphism theorems

For $X \subset \mathbb {Z}^m$ we set, for each $q \in \{0, \ldots , n \}$,

$$\begin{aligned} \mathscr {C}^\infty _X (\Omega ; \underline{\Lambda }^{q}) \doteq \{ f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q} ) \ ; \ {{\hat{f}}}_\xi = 0, \ \forall \xi \in \mathbb {Z}^m {\setminus } X \}, \end{aligned}$$

and for $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$, we consider its projection on $\mathscr {C}^\infty _X (\Omega ; \underline{\Lambda }^{q})$:

$$\begin{aligned} f_X \doteq \frac{1}{(2\pi )^m} \sum _{\xi \in X} e^{ix \xi } \wedge {{\hat{f}}}_\xi . \end{aligned}$$

It follows that $f = f_X + f_{\mathbb {Z}^m {\setminus } X}$ and hence

$$\begin{aligned} \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q} ) = \mathscr {C}^\infty _X (\Omega ; \underline{\Lambda }^{q}) \oplus \mathscr {C}^\infty _{\mathbb {Z}^m {\setminus } X} (\Omega ; \underline{\Lambda }^{q}). \end{aligned}$$

Notice moreover that

$$\begin{aligned} \textrm{d}' \mathscr {C}^\infty _X (\Omega ; \underline{\Lambda }^{q}) \subset \mathscr {C}^\infty _X (\Omega ; \underline{\Lambda }^{q + 1}) \end{aligned}$$

and that

$$\begin{aligned} \textrm{d}' u = f \Longleftrightarrow \textrm{d}' u_X = f_X \quad \text {and}\quad \textrm{d}' u_{\mathbb {Z}^m {\setminus } X} = f_{\mathbb {Z}^m {\setminus } X}. \end{aligned}$$

It thus makes sense to define

$$\begin{aligned} H^{q}_{\textrm{d}'}(\mathscr {C}^\infty _X(\Omega )) \doteq \frac{ \ker \{ \textrm{d}': \mathscr {C}^\infty _X(\Omega ; \underline{\Lambda }^{q}) \longrightarrow \mathscr {C}^\infty _X(\Omega ; \underline{\Lambda }^{q + 1}) \} }{ {{\,\mathrm{\textrm{ran}}\,}}\{ \textrm{d}': \mathscr {C}^\infty _X(\Omega ; \underline{\Lambda }^{q - 1}) \longrightarrow \mathscr {C}^\infty _X(\Omega ; \underline{\Lambda }^{q}) \} }, \end{aligned}$$

hence, we have

$$\begin{aligned} H^{q}_{\textrm{d}'}(\mathscr {C}^\infty (\Omega )) \cong H^{q}_{\textrm{d}'}(\mathscr {C}^\infty _X(\Omega )) \oplus H^{q}_{\textrm{d}'}(\mathscr {C}^\infty _{\mathbb {Z}^m {\setminus } X}(\Omega )). \end{aligned}$$

We denote by $\pi : {\tilde{M}} \rightarrow M$ the universal covering of M. For each $k \in \{1, \ldots , m\}$ there exists $\psi _k \in \mathscr {C}^\infty ({\tilde{M}}; \mathbb {R})$ such that $\textrm{d}\psi _k = \pi ^* \omega _k$ on $\tilde{M}$ (since the latter form is exact). More generally, for $\xi \in \mathbb {Z}^m$ we set

$$\begin{aligned} \pmb {\psi }_\xi \doteq \sum _{k = 1}^m \xi _k \psi _k \in \mathscr {C}^\infty ({\tilde{M}}; \mathbb {R}), \end{aligned}$$

(5.1)

hence $\textrm{d}\pmb {\psi }_\xi = \pi ^* (\xi \cdot \pmb {\omega })$. In an open set $U \subset M$ in which $\pi _U: \tilde{U} \rightarrow U$ is diffeomorphism ($\tilde{U} \subset {\tilde{M}}$ being another open set) we have

$$\begin{aligned} \textrm{d}e^{i\pmb {\psi }_\xi \circ \pi _U^{-1}} = i e^{i\pmb {\psi }_\xi \circ \pi _U^{-1}} (\xi \cdot \pmb {\omega }) \quad \hbox {on }\ U. \end{aligned}$$

If $\xi \in \Gamma _{\pmb {\omega }}$, then $\xi \cdot \pmb {\omega }$ is integral, hence by [3, Lemma 2.3], if $P,Q \in {\tilde{M}}$ satisfy $\pi (P) = \pi (Q)$ then $\pmb {\psi }_\xi (P) - \pmb {\psi }_\xi (Q) \in 2 \pi \mathbb {Z}$, we can define for every $\xi \in \Gamma _{\pmb {\omega }}$ a smooth function

$$\begin{aligned} M \ni t \longmapsto e^{i \pmb {\psi }_\xi \circ \pi ^{-1}(t)}, \end{aligned}$$

(5.2)

even though $\pi ^{-1}$ is not a function. It follows that

$$\begin{aligned} e^{i \pmb {\psi }_\xi \circ \pi ^{-1}} \in \mathscr {S}_{(- \xi \cdot \pmb {\omega })} (M) {\setminus } \{0\}, \quad \forall \xi \in \Gamma _{\pmb {\omega }}. \end{aligned}$$

We define a map $\Psi : {\tilde{M}} \times \mathbb {T}^m \rightarrow {\tilde{M}} \times \mathbb {T}^m$ by

$$\begin{aligned} \Psi ({\tilde{t}}, x_1, \ldots , x_m) \doteq ({\tilde{t}}, x_1 - \psi _1(\tilde{t}), \ldots , x_m - \psi _m({\tilde{t}}) ), \end{aligned}$$

in which the sum $x_k - \psi _k(\tilde{t})$ takes place in $\mathbb {R}/(2\pi \mathbb {Z})$. In particular, $\Psi$ is a diffeomorphism with inverse

$$\begin{aligned} \Psi ^{-1} ({\tilde{t}}, x_1, \ldots , x_m ) = ({\tilde{t}}, x_1 + \psi _1({\tilde{t}}), \ldots , x_m + \psi _m({\tilde{t}}) ). \end{aligned}$$

Proposition 5.1

Assume $\omega _1, \ldots , \omega _m$ real-valued and let $f \in \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}} (\Omega ; \underline{\Lambda }^{q})$. Then

$$\begin{aligned} \Theta (f) \doteq \frac{1}{(2\pi )^m} \sum _{\xi \in \Gamma _{\pmb {\omega }}} e^{ix \xi - i \pmb {\psi }_\xi \circ \pi ^{-1}} {{\hat{f}}}_\xi \end{aligned}$$

(5.3)

defines an element in $\mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}} (\Omega ; \underline{\Lambda }^{q})$. The map

$$\begin{aligned} \Theta : \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}} (\Omega ; \underline{\Lambda }^{q}) \longrightarrow \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}} (\Omega ; \underline{\Lambda }^{q}) \end{aligned}$$

is a linear isomorphism and satisfies $\textrm{d}' \circ \Theta = \textrm{d}_t \circ \Theta$.

Proof

Writing the partial Fourier series of f, we have that

$$\begin{aligned} f(t,x) = \frac{1}{(2\pi )^m} \sum _{\xi \in \Gamma _{\pmb {\omega }}} e^{ix \xi } {\hat{f}}_\xi (t). \end{aligned}$$

We define $F \doteq (\pi \times \textrm{id}_{\mathbb {T}^m})^* f$, which is a smooth q-form on $\tilde{M} \times \mathbb {T}^m$. By continuity of the pullback map, it follows that

$$\begin{aligned} F(\tilde{t}, x) = \frac{1}{(2\pi )^m} \sum _{\xi \in \Gamma _{\pmb {\omega }}} e^{ix \xi } (\pi ^* {\hat{f}}_\xi )(\tilde{t}), \end{aligned}$$

with convergence in the space of smooth q-forms on $\tilde{M} \times \mathbb {T}^m$. If we change coordinates using the diffeomorphism $\Psi$, we obtain

$$\begin{aligned} (\Psi ^* F)(\tilde{t}, x) = \frac{1}{(2\pi )^m} \sum _{\xi \in \Gamma _{\pmb {\omega }}} e^{i(x \xi - \pmb {\psi }_\xi (\tilde{t}))} (\pi ^* {\hat{f}}_\xi )(\tilde{t}), \end{aligned}$$

which we will show that descends to $M \times \mathbb {T}^m$ as $\Theta (f)$. The sequence of truncated sums

$$\begin{aligned} f_\nu (t) \doteq \frac{1}{(2\pi )^m} \sum _{\begin{array}{c} \xi \in \Gamma _{\pmb {\omega }} \\ |\xi | \le \nu \end{array}} e^{ix \xi } {\hat{f}}_\xi (t), \quad \nu \in \mathbb {N}, \end{aligned}$$

which approximates f in $\mathscr {C}^\infty (\Omega ; \underline{\Lambda }^q)$, certainly satisfy

$$\begin{aligned} \Psi ^* (\pi \times \textrm{id}_{\mathbb {T}^m})^* f_\nu \longrightarrow \Psi ^* F \quad \hbox { on }\ \tilde{M} \times \mathbb {T}^m \end{aligned}$$

whereas

$$\begin{aligned} g_\nu (t) \doteq \frac{1}{(2\pi )^m} \sum _{\begin{array}{c} \xi \in \Gamma _{\pmb {\omega }} \\ |\xi | \le \nu \end{array}} e^{ix \xi - i \pmb {\psi }_\xi \circ \pi ^{-1}(t)} {{\hat{f}}}_\xi (t), \quad \nu \in \mathbb {N}, \end{aligned}$$

satisfy $(\pi \times \textrm{id}_{\mathbb {T}^m})^* g_\nu = \Psi ^* (\pi \times \textrm{id}_{\mathbb {T}^m})^* f_\nu$ clearly, hence

$$\begin{aligned} (\pi \times \textrm{id}_{\mathbb {T}^m})^* g_\nu \longrightarrow \Psi ^* F \quad \hbox { on }\ \tilde{M} \times \mathbb {T}^m. \end{aligned}$$

(5.4)

To prove convergence of $\{ g_\nu \}_{\nu \in \mathbb {N}}$ in $\mathscr {C}^\infty (\Omega ; \underline{\Lambda }^q)$, it suffices to check it on $U \times \mathbb {T}^m$ for a suitably small open set $U \subset M$: its limit will then be automatically a globally defined q-form $\Theta (f)$ on $M \times \mathbb {T}^m$, and can be easily seen to be a section of $\underline{\Lambda }^q$. This is the case if, say, there exists an open set $\tilde{U} \subset \tilde{M}$ such that $\pi _U: \tilde{U} \rightarrow U$ is a diffeomorphism. Since the previous convergence (5.4) takes place in $\tilde{U} \times \mathbb {T}^m$ as well by restriction, in which $\pi _U \times \textrm{id}_{\mathbb {T}^m}$ is invertible, we conclude that $\{ g_\nu \}_{\nu \in \mathbb {N}}$ converges in $U \times \mathbb {T}^m$, proving our claim.

It is clear that (5.3) holds (as $\Theta (f)$ is by definition the limit of $\{ g_\nu \}_{\nu \in \mathbb {N}}$), hence of course $\Theta (f) \in \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}} (\Omega ; \underline{\Lambda }^{q})$: by continuity of each map $\mathcal {F}_\xi$, Eq. (5.3) gives its Fourier coefficients explicitly. Moreover:

$$\begin{aligned} (2\pi )^m \textrm{d}' \Theta (f)&= \left( \textrm{d}_t + \sum _{k = 1}^m \omega _k \wedge \frac{\partial }{\partial x_k} \right) \sum _{\xi \in \Gamma _{\pmb {\omega }}} e^{ix \xi -i \pmb {\psi }_\xi \circ \pi ^{-1}} {{\hat{f}}}_\xi \\&= \sum _{\xi \in \Gamma _{\pmb {\omega }}} \left( e^{ix \xi } \textrm{d}_t( e^{ -i \pmb {\psi }_\xi \circ \pi ^{-1}} \hat{ f_\xi }) + \sum _{k = 1}^m \frac{\partial e^{ix \xi }}{\partial x_k} \omega _k\wedge ( e^{-i \pmb {\psi }_\xi \circ \pi ^{-1}} {{\hat{f}}}_\xi ) \right) \\&= \sum _{\xi \in \Gamma _{\pmb {\omega }}} \left( e^{ix \xi - i \pmb {\psi }_\xi \circ \pi ^{-1}} \left( \textrm{d}_t {{\hat{f}}}_\xi - i (\xi \cdot \pmb {\omega }) \wedge {{\hat{f}}}_\xi \right) + i (\xi \cdot \pmb {\omega }) \wedge e^{ix \xi - i \pmb {\psi }_\xi \circ \pi ^{-1}} {{\hat{f}}}_\xi \right) \\&= \sum _{\xi \in \Gamma _{\pmb {\omega }}} e^{ix \xi - i \pmb {\psi }_\xi \circ \pi ^{-1}} \widehat{\textrm{d}_t f_\xi } \\&= (2\pi )^m \Theta (\textrm{d}_t f), \end{aligned}$$

that is: $\Theta ^{-1} \circ \textrm{d}' \circ \Theta = \textrm{d}_t$ on $\mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}} (\Omega ; \underline{\Lambda }^{q})$. $\square$

A similar calculation shows that, if

$$\begin{aligned} \Theta _\xi : \mathscr {C}^\infty (M; \Lambda ^q) \longrightarrow \mathscr {C}^\infty (M; \Lambda ^q), \quad \Theta _\xi (h) \doteq e^{i \pmb {\psi }_\xi \circ \pi ^{-1}} h, \end{aligned}$$

then $\Theta _\xi ^{-1} \circ \textrm{d}'_\xi \circ \Theta _\xi = \textrm{d}$, whatever $\xi \in \Gamma _{\pmb {\omega }}$.

Theorem 5.2

Assume $\omega _1, \ldots , \omega _m$ real-valued. Then, $\textrm{d}': \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q}) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q + 1})$ has closed range if and only if given a formally solvable $f \in \mathscr {C}^\infty _{\mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}} (\Omega ; \underline{\Lambda }^{q + 1})$, there exists $u \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ such that $\textrm{d}'u = f$.

Proof

The direct implication is granted by Corollary 4.8; we prove the converse. Let $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q + 1})$ be such that for each $\xi \in \mathbb {Z}^m$ there exists $u_\xi \in \mathscr {C}^\infty (M; \Lambda ^q)$ satisfying $\textrm{d}'_\xi u_\xi = {{\hat{f}}}_\xi$. Write $f = f_{\Gamma _{\pmb {\omega }}} + f_{\mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}}$ and notice that both $f_{\Gamma _{\pmb {\omega }}}$ and $f_{\mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}}$ are also formally solvable. By hypothesis, there exists $v \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ such that $\textrm{d}' v = f_{\mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}}$: we must solve $\textrm{d}' u = f_{\Gamma _{\pmb {\omega }}}$.

We claim that $g \doteq \Theta ^{-1} f_{\Gamma _{\pmb {\omega }}} \in \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}} (\Omega ; \underline{\Lambda }^{q + 1} )$ is formally solvable w.r.t. $\textrm{d}_t$ in $\Omega$. Indeed, ${{\hat{g}}}_\xi = 0$ for $\xi \in \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$, while for $\xi \in \Gamma _{\pmb {\omega }}$, we have

$$\begin{aligned} \textrm{d}'_\xi u_\xi = {\hat{f}}_\xi \Longrightarrow \textrm{d}_t \Theta ^{-1}_\xi u_\xi = ( \Theta _\xi ^{-1} \textrm{d}_\xi ' \Theta _\xi ) \Theta ^{-1}_\xi u_\xi = \Theta _\xi ^{-1} {{\hat{f}}}_\xi = \mathcal {F}_\xi (\Theta ^{-1} f_{\Gamma _{\pmb {\omega }}}) = {{\hat{g}}}_\xi . \end{aligned}$$

But global solvability of $\textrm{d}_t$ is a general fact—see Lemma 5.3 below—which will also play a role later on. It then follows from Corollary 4.8 applied to $\textrm{d}_t$ that there exists $w \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ such that $\textrm{d}_t w = g$. Note that we can replace w with $w_{\Gamma _{\pmb {\omega }}}$, if necessary, and assume that $w \in \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\Omega ; \underline{\Lambda }^{q})$. If we set $u \doteq \Theta w \in \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\Omega ; \underline{\Lambda }^{q})$ then

$$\begin{aligned} \textrm{d}' u = \textrm{d}' \Theta w = \Theta \textrm{d}_t w = \Theta g = f_{\Gamma _{\pmb {\omega }}} \end{aligned}$$

and we are done. $\square$

Lemma 5.3

The map $\textrm{d}_t: \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q}) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q + 1})$ has closed range.

Proof

We endow M with a Riemannian metric and let $\Delta \doteq \textrm{d}\textrm{d}^* + \textrm{d}^* \textrm{d}$ be the Laplace–Beltrami operator acting on forms on M; it is elliptic of order 2 and therefore satisfies elliptic estimates: given $k \in \mathbb {Z}_+$ there exists $C_k > 0$ such that

$$\begin{aligned} \Vert \psi \Vert _{\mathscr {H}^{k+2} (M; \Lambda ^q)} \le C_k \left( \Vert \Delta \psi \Vert _{\mathscr {H}^{k} (M; \Lambda ^q)} + \Vert \psi \Vert _{\mathscr {H}^{k} (M; \Lambda ^q)} \right) , \quad \forall \psi \in \mathscr {H}^{k+2} (M; \Lambda ^q), \end{aligned}$$

where $\mathscr {H}^k$ are Sobolev spaces. A standard argument shows that

$$\begin{aligned} \Vert \psi \Vert _{\mathscr {H}^{k+2} (M; \Lambda ^q)} \le C_k \Vert \Delta \psi \Vert _{\mathscr {H}^{k} (M; \Lambda ^q)}, \quad {\forall \psi \in \mathscr {H}^{k+2} (M; \Lambda ^q), L^2\hbox {- orthogonal to} \ker \Delta }. \end{aligned}$$

The following orthogonal decomposition w.r.t. the $L^2(M)$ metric is also well-known:

$$\begin{aligned} \mathscr {C}^\infty (M; \Lambda ^q) = \ker \Delta \oplus {{\,\mathrm{\textrm{ran}}\,}}\textrm{d}\oplus {{\,\mathrm{\textrm{ran}}\,}}\textrm{d}^* \end{aligned}$$

(5.5)

(all operators acting on smooth forms), hence in particular, for $\psi \in {{\,\mathrm{\textrm{ran}}\,}}\textrm{d}^*$,

$$\begin{aligned} \Vert \psi \Vert _{\mathscr {H}^{k + 2} (M; \Lambda ^q)} \le C_k \Vert \textrm{d}^* \textrm{d}\psi \Vert _{\mathscr {H}^{k} (M; \Lambda ^q)} \le C_k' \Vert \textrm{d}\psi \Vert _{\mathscr {H}^{k - 1} (M; \Lambda ^{q + 1})} \end{aligned}$$

since in that case $\Delta \psi = \textrm{d}^* \textrm{d}\psi$. We conclude that, for each $k \in \mathbb {Z}_+$, there exist $c_k > 0$ and $j \in \mathbb {Z}_+$ such that

$$\begin{aligned} \Vert \psi \Vert _{\mathscr {H}^{k} (M; \Lambda ^q)} \le c_k \Vert \textrm{d}\psi \Vert _{\mathscr {H}^{j} (M; \Lambda ^{q + 1})}, \quad \forall \psi \in {{\,\mathrm{\textrm{ran}}\,}}\textrm{d}^*. \end{aligned}$$

(5.6)

Now, we obtain our conclusion by means of Theorem 2.2. Let $u \in \mathscr {D}'(\Omega ; \underline{\Lambda }^{q})$ be such that $\textrm{d}_t u \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q + 1})$. Then, for every $\xi \in \mathbb {Z}^m$, we have that $\textrm{d}{\hat{u}}_\xi \in \mathscr {C}^\infty (M; \Lambda ^{q + 1})$ by Corollary 2.9, hence, by Theorem 3.1, there exists $v_\xi \in \mathscr {C}^\infty (M; \Lambda ^q)$ such that $\textrm{d}v_\xi = \textrm{d}{\hat{u}}_\xi$; thanks to (5.5) we can further assume that $v_\xi \in {{\,\mathrm{\textrm{ran}}\,}}\textrm{d}^*$. By (5.6), for each $k \in \mathbb {Z}_+$, there exist $c_k > 0$ and $j \in \mathbb {Z}_+$ such that

$$\begin{aligned} \Vert v_\xi \Vert _{\mathscr {H}^{k}(M; \Lambda ^q)} \le c_k \Vert \textrm{d}v_\xi \Vert _{\mathscr {H}^j(M; \Lambda ^{q + 1})}, \quad \forall \xi \in \mathbb {Z}^m. \end{aligned}$$

Moreover, since $\textrm{d}_t u$ is smooth and $\mathcal {F}_\xi (\textrm{d}_t u) = \textrm{d}{\hat{u}}_\xi$, for every $j \in \mathbb {Z}_+$ and $s > 0$, there exists a constant $A_{j,s} > 0$ such that

$$\begin{aligned} \Vert \textrm{d}{\hat{u}}_\xi \Vert _{\mathscr {H}^j(M; \Lambda ^{q + 1})} \le A_{j,s} (1 + |\xi |^2)^{-s}, \quad \forall \xi \in \mathbb {Z}^m, \end{aligned}$$

and so, for $c'_{k,s} \doteq c_k A_{j,s}$, we have

$$\begin{aligned} \Vert v_\xi \Vert _{\mathscr {H}^{k}(M; \Lambda ^{q})} \le c'_{k,s} (1 + |\xi |^2)^{-s}, \quad \forall \xi \in \mathbb {Z}^m. \end{aligned}$$

(5.7)

The latter ensures that the series

$$\begin{aligned} v \doteq \frac{1}{(2 \pi )^m} \sum _{\xi \in \mathbb {Z}^m} e^{ix \xi } \wedge v_\xi \end{aligned}$$

converges in $L^2(\Omega ; \underline{\Lambda }^{q})$ since

$$\begin{aligned} \frac{1}{(2 \pi )^m} \sum _{\xi } \int _\Omega \Vert e^{i x \xi } v_\xi (t) \Vert _{\Lambda ^q_t}^2 \textrm{d}V(t,x)&= \sum _{\xi } \int _M \Vert v_\xi (t) \Vert _{\Lambda ^q_t}^2 \textrm{d}V_M(t) \\&\le (c'_{0, 2m})^2 \sum _{\xi } (1 + |\xi |^2)^{-2m} < \infty . \end{aligned}$$

Moreover, for each $\xi \in \mathbb {Z}^m$, we have that ${\hat{v}}_\xi = v_\xi$, hence $\textrm{d}{\hat{v}}_\xi = \textrm{d}{\hat{u}}_\xi$, so $\textrm{d}_t v = \textrm{d}_t u$. Estimates (5.7) further ensure that $v \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$. $\square$

Remark 5.4

We have proved, in Theorem 5.2, that $\textrm{d}'$ is always $\Gamma _{\pmb {\omega }}$-globally solvable, that is, if $f \in \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}} (\Omega ; \underline{\Lambda }^{q + 1})$ is formally solvable, then there exists $u \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ such that $\textrm{d}'u = f$, with no further hypotheses.

5.1.1 Reduction in cohomology

Assume $\omega _1, \ldots , \omega _m$ real-valued. The map $\Theta$ descends to a linear isomorphism

$$\begin{aligned} H^{q}_{\textrm{d}'}(\mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\Omega )) \cong H^{q}_{\textrm{d}_t}(\mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\Omega )) \end{aligned}$$

thanks to its properties deduced above. Now, we provide a more detailed description of those spaces, for which we introduce some notation. Take $\xi ^{(1)}, \ldots , \xi ^{(r)}$ a basis of $\Gamma _{\pmb {\omega }}$ as a $\mathbb {Z}$-module⁴, thus, for each $\xi \in \Gamma _{\pmb {\omega }}$, there exists a unique $\eta \in \mathbb {Z}^{r}$ such that

$$\begin{aligned} \xi = \eta _1 \xi ^{(1)} + \cdots + \eta _r \xi ^{(r)} \doteq \eta \cdot \xi ^{0} \end{aligned}$$

in which $\xi ^{0} \doteq (\xi ^{(1)}, \ldots , \xi ^{(r)})$. We also define a smooth map $\theta : \mathbb {T}^m \rightarrow \mathbb {T}^r$ by

$$\begin{aligned} \theta (x_1, \ldots , x_m) \doteq \left( x \cdot \xi ^{(1)}, \ldots , x \cdot \xi ^{(r)} \right) . \end{aligned}$$

Note that

$$\begin{aligned} \sum _{j = 1}^r (x \cdot \xi ^{(j)}) \eta _j = x_1 \Big ( \sum _{j = 1}^r \xi _1^{(j)} \eta _j \Big ) + \cdots + x_m \Big ( \sum _{j = 1}^r \xi _m^{(j)} \eta _j \Big ) = x \cdot \xi \end{aligned}$$

thus

$$\begin{aligned} e^{i \theta (x) \cdot \eta } = \prod _{j = 1}^r (e^{i x \cdot \xi ^{(j)}} )^{\eta _j} = e^{i x \cdot (\eta \cdot \xi ^0)}= e^{i x \xi }. \end{aligned}$$

Proposition 5.5

Assume $\omega _1, \ldots , \omega _m$ real-valued. Then the pullback $\theta ^*: \mathscr {C}^\infty (\mathbb {T}^r) \rightarrow \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\mathbb {T}^m)$ is a linear isomorphism.

Proof

We write an $f \in \mathscr {C}^\infty (\mathbb {T}^r)$ as

$$\begin{aligned} f = \frac{1}{(2 \pi )^r} \sum _{\eta \in \mathbb {Z}^r} e^{i y \eta } {\hat{f}}_{\eta } \end{aligned}$$

with convergence in $\mathscr {C}^\infty (\mathbb {T}^r)$. Hence

$$\begin{aligned} \theta ^* f = \frac{1}{(2 \pi )^r} \sum _{\eta \in \mathbb {Z}^r} e^{i \theta (x) \cdot \eta } {\hat{f}}_{\eta } = \frac{1}{(2 \pi )^r} \sum _{\eta \in \mathbb {Z}^r} e^{i x (\eta \cdot \xi ^0)} {\hat{f}}_{\eta } \end{aligned}$$

meaning that

$$\begin{aligned} \mathcal {F}_\xi (\theta ^* f) = {\left\{ \begin{array}{ll} (2 \pi )^{m - r} {\hat{f}}_{\eta }, &{}\hbox { if}\ \xi = \eta \cdot \xi ^0 \in \Gamma _{\pmb {\omega }}; \\ 0, &{}\hbox { if}\ \xi \notin \Gamma _{\pmb {\omega }}. \end{array}\right. } \end{aligned}$$

In particular, $\theta ^* f \in \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\mathbb {T}^m)$. Moreover, $\theta ^*$ is injective since $\theta ^* f = 0$ implies that ${\hat{f}}_{\eta } = 0$ for all $\eta \in \mathbb {Z}^r$. As for surjectivity, given $g \in \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\mathbb {T}^m)$, let

$$\begin{aligned} f \doteq \frac{1}{(2 \pi )^r} \sum _{\eta \in \mathbb {Z}^r} e^{i y \eta } {\hat{g}}_{\eta \cdot \xi ^0} \in \mathscr {C}^\infty (\mathbb {T}^r) \end{aligned}$$

which clearly satisfies $\theta ^* f = g$.

Note that $|\xi |$ and $|\eta |$ are comparable, so a series converges in $\mathscr {C}^\infty (\mathbb {T}^r)$ if and only if its image by $\theta ^*$ converges in $\mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\mathbb {T}^m)$. Indeed, we can complete $\xi ^{0}$ to a basis of $\mathbb {R}^m$, so we can write $A = (\xi ^{(1)}, \ldots , \xi ^{(r)}, \xi ^{(r+1)}, \ldots , \xi ^{(m)})$, in which A is a matrix whose j-th column equals $\xi ^{(j)}$. It follows that multiplication of A by $(\eta , 0) \in \mathbb {Z}^{m}$ yields

$$\begin{aligned} (\eta , 0) \cdot A = \eta \cdot \xi ^{0} = \xi . \end{aligned}$$

Since A is invertible, it follows that $c |\eta | \le |\xi | \le C |\eta |$ for some constants $c, C > 0$. $\square$

Now given $q \in \{0, \ldots , n\}$, for any $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ and $x \in \mathbb {T}^m$, we define $f(x) \in \mathscr {C}^\infty (M; \Lambda ^q)$ by “fixing the x-variable”, i.e. $f(t,x) = f(x)(t)$ for every $t \in M$. As such, if $\textrm{d}_t f = 0$, then $\textrm{d}f(x) = 0$, and, if we pick closed forms $\tau _1, \ldots , \tau _{b_q} \in \mathscr {C}^\infty (M; \Lambda ^q)$ such that $[\tau _1], \ldots , [\tau _{b_q}]$ is a basis of $H^q_{\textrm{dR}}(M)$, then for each $x \in \mathbb {T}^m$ we write

$$\begin{aligned}{}[f(x)] = \sum _{\ell = 1}^{b_q} a_\ell (x) [\tau _\ell ] \quad \text {in } H^q_{\text {dR}}(M) \end{aligned}$$

for some uniquely determined coefficients $a_1(x), \ldots , a_{b_q}(x)$. Things can be arranged so that $x \in \mathbb {T}^m \mapsto a_\ell (x) \in \mathbb {C}$ are all smooth.

Indeed, by endowing M with a Riemannian metric, we may pick $\tau _1, \ldots , \tau _{b_q}$ as a basis of $\ker \Delta$, the space of harmonic q-forms on M, which is orthonormal w.r.t. the $L^2$ inner product on $\mathscr {C}^\infty (M; \Lambda ^q)$. Then there exists $u_x \in \mathscr {C}^\infty (M; \Lambda ^{q - 1})$ such that

$$\begin{aligned} f(x) = \sum _{\ell = 1}^{b_q} a_\ell (x) \tau _\ell + \textrm{d}_t u_x, \end{aligned}$$

thus realizing the last sum as the orthogonal projection of f(x) onto $\ker \Delta$ (see (5.5)):

$$\begin{aligned} a_\ell (x) = \langle f(x), \tau _\ell \rangle _{L^2(M; \Lambda ^q)} = \int _M \langle f(t,x), \tau _\ell (t) \rangle _{\Lambda ^q_t} \ \textrm{d}V_M(t), \quad \ell \in \{1, \ldots , b_q\},\nonumber \\ \end{aligned}$$

(5.8)

are therefore smooth w.r.t. x.

Notice also that $a_1(x), \ldots , a_{b_q}(x)$ depend only on the cohomology class of f: if $f^\bullet \doteq f + \textrm{d}_t v$ for some $v \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1})$ then

$$\begin{aligned} f^\bullet (x) = f(x) + \textrm{d}_t v(x) = \sum _{\ell = 1}^{b_q} a_\ell (x) \tau _\ell + \textrm{d}_t (u_x + v(x)). \end{aligned}$$

It follows from (5.8) that, for every $\xi \in \mathbb {Z}^m$,

$$\begin{aligned} \mathcal {F}_\xi (a_\ell ) = \int _{\mathbb {T}^m} e^{-ix \xi } \int _M \langle f(t,x), \tau _\ell (t) \rangle _{\Lambda ^q_t} \ \textrm{d}V_M(t) \textrm{d}x = \langle {\hat{f}}_\xi , \tau _\ell \rangle _{L^2(M; \Lambda ^q)}. \end{aligned}$$

(5.9)

If we further assume that $f \in \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}} (\Omega ; \underline{\Lambda }^{q})$, then by (5.9) we have $\mathcal {F}_\xi (a_\ell ) = 0$ whenever $\xi \notin \Gamma _{\pmb {\omega }}$, that is, $a_\ell \in \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}} (\mathbb {T}^m)$, so, by Proposition 5.5, there is a unique $a_\ell ^\bullet \in \mathscr {C}^\infty (\mathbb {T}^r)$ such that $a_\ell = \theta ^* (a_\ell ^\bullet )$. We define

$$\begin{aligned} {\textsf{T}}: H^{q}_{\textrm{d}_t}(\mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\Omega )) \longrightarrow \mathscr {C}^\infty (\mathbb {T}^r) \otimes H^q_{\textrm{dR}}(M) \end{aligned}$$

by

$$\begin{aligned} {\textsf{T}}([f]) \doteq \sum _{\ell = 1}^{b_q} a_\ell ^\bullet \otimes [\tau _\ell ]. \end{aligned}$$

Theorem 5.6

Assume $\omega _1, \ldots , \omega _m$ real-valued. Then ${\textsf{T}}$ is a linear isomorphism.

Proof

The surjectivity is obvious; given $a_1^\bullet , \ldots , a_{b_q}^\bullet \in \mathscr {C}^\infty (\mathbb {T}^r)$, we have

$$\begin{aligned} f \doteq \sum _{\ell = 1}^{b_q} (\theta ^* a_\ell ^\bullet ) \tau _\ell \Longrightarrow {\textsf{T}}([f]) = \sum _{\ell = 1}^{b_q} a_\ell ^\bullet \otimes [\tau _\ell ]. \end{aligned}$$

As for injectivity, if a $\textrm{d}_t$-closed $f \in \mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\Omega ; \underline{\Lambda }^{q})$ is such that ${\textsf{T}}([f]) = 0$, then $a_1 = \cdots = a_{b_q} = 0$. Since $\textrm{d}_t f = 0$, it follows that $\textrm{d}{\hat{f}}_\xi = 0$ for every $\xi \in \Gamma _{\pmb {\omega }}$, and so we can find $a^\xi _\ell \in \mathbb {C}$ and $u_{\xi }\in \mathscr {C}^\infty (M; \Lambda ^{q - 1})$ such that

$$\begin{aligned} {\hat{f}}_\xi = \sum _{\ell = 1}^{b_q} a^{\xi }_\ell \tau _\ell + \textrm{d}u_{\xi }, \quad \forall \xi \in \Gamma _{\pmb {\omega }}. \end{aligned}$$

Note that for $\xi \notin \Gamma _{\pmb {\omega }}$ we have ${\hat{f}}_\xi = 0$. By (5.9), we have, for each $\ell \in \{1, \ldots , b_q\}$ and $\xi \in \Gamma _{\pmb {\omega }}$, using that $\tau _\ell \in \ker \Delta = \ker \textrm{d}\cap \ker \textrm{d}^*$:

$$\begin{aligned} \mathcal {F}_\xi (a_\ell ) = a^{\xi }_\ell + \langle \textrm{d}u_{\xi }, \tau _\ell \rangle _{L^2(M; \Lambda ^q)} = a^{\xi }_\ell . \end{aligned}$$

By the assumption that $a_\ell = 0$ for every $\ell \in \{1, \ldots , b_q\}$, we reach the conclusion that ${\hat{f}}_\xi$ is exact for every $\xi \in \mathbb {Z}^m$, that is, f is formally $\textrm{d}_t$-solvable. By Corollary 4.8 and Lemma 5.3, we conclude that f is $\textrm{d}_t$-exact, i.e. $[f] = 0$ in $H^{q}_{\textrm{d}_t}(\mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\Omega ))$. $\square$

5.1.2 Summary of the section

When $\omega _1, \ldots , \omega _m$ are real, we always have linear isomorphisms

$$\begin{aligned} H^{q}_{\textrm{d}'}(\mathscr {C}^\infty (\Omega )) \cong H^{q}_{\textrm{d}'}(\mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\Omega )) \oplus H^{q}_{\textrm{d}'}(\mathscr {C}^\infty _{\mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}}(\Omega )) \end{aligned}$$

and

$$\begin{aligned} H^{q}_{\textrm{d}'}(\mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\Omega )) \cong H^{q}_{\textrm{d}_t}(\mathscr {C}^\infty _{\Gamma _{\pmb {\omega }}}(\Omega )) \cong \mathscr {C}^\infty (\mathbb {T}^r) \otimes H^q_{\textrm{dR}}(M); \end{aligned}$$

(5.10)

recall that r is the dimension of $\Gamma _{\pmb {\omega }}$ as a $\mathbb {Z}$-module. We finish this section with the following theorem, which introduces a condition that will appear later in a more general situation, and that we will prove to hold in several cases (for notation, see (2.8)).

Theorem 5.7

Assume $\omega _1, \ldots , \omega _m$ real-valued. Suppose that $\textrm{d}': \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1}) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ has closed range and

$$\begin{aligned} H_\xi ^q (\mathscr {C}^\infty (M)) = 0, \quad \forall \xi \in \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}. \end{aligned}$$

(5.11)

Then

$$\begin{aligned} H^{q}_{\textrm{d}'}(\mathscr {C}^\infty _{\mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}}(\Omega )) = \{0\}. \end{aligned}$$

In particular, in that case (1.5) holds.

Proof

Let $f \in \mathscr {C}^\infty _{\mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}}(\Omega ; \underline{\Lambda }^{q})$ be such that $\textrm{d}' f = 0$. Hence $\textrm{d}'_\xi {\hat{f}}_\xi = 0$, and by (5.11), there exists $u_\xi \in \mathscr {C}^\infty (M; \Lambda ^{q - 1})$ such that $\textrm{d}'_\xi u_\xi = {\hat{f}}_\xi$ for every $\xi \in \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$. Therefore f is formally solvable, and by Theorem 5.2, there exists $u \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1})$ such that $\textrm{d}' u = f$. Replacing u by $u_{\mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}}$ yields the desired result. $\square$

5.2 Global solvability in the first degree

Theorem 5.2 tells us that the obstruction to global solvability of $\textrm{d}'$ is encoded in the frequencies $\xi \in \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$. This fact motivates us to consider the following Diophantine condition. Our approach in this section follows and adapts ideas from [3], hence we omit some proofs.

Definition 5.8

A collection $\pmb {\omega } = (\omega _1, \ldots , \omega _m)$ of real closed 1-forms on M is said to be strongly simultaneously approximable if there exist a sequence of closed integral 1-forms $\{ \beta _\nu \}_{\nu \in \mathbb {N}} \subset \mathscr {C}^\infty (M; \Lambda ^1)$ and $\{ \xi _\nu \}_{\nu \in \mathbb {N}} \subset \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$ such that $|\xi _\nu | \rightarrow \infty$ and

$$\begin{aligned} \{ | \xi _\nu |^\nu (\xi _\nu \cdot \pmb {\omega } - \beta _\nu ) \}_{\nu \in \mathbb {N}} \quad \hbox {is bounded in }\ \mathscr {C}^\infty (M; \Lambda ^1). \end{aligned}$$

(5.12)

Otherwise, it is said to be weakly non-simultaneously approximable.

Such notions depend only on the classes $[\omega _1], \ldots , [\omega _m] \in H^1_{\textrm{dR}} (M)$. Indeed, if for each $k \in \{1, \ldots , m\}$ we have $\omega _k^\bullet \doteq \omega _k + \textrm{d}\gamma _k$ for some $\gamma _k \in \mathscr {C}^\infty (M; \mathbb {R})$ then

$$\begin{aligned} \xi \cdot \pmb {\omega }^\bullet = \xi \cdot \pmb {\omega } + \xi \cdot \textrm{d}\pmb {\gamma }, \quad \forall \xi \in \mathbb {Z}^m, \end{aligned}$$

in which $\pmb {\omega }^\bullet \doteq (\omega _1^\bullet , \ldots , \omega _m^\bullet )$ and $\pmb {\gamma } \doteq (\gamma _1, \ldots , \gamma _m)$: by integrating both sides against an arbitrary 1-cycle, it follows that $\xi \cdot \pmb {\omega }^\bullet$ is integral if and only if so is $\xi \cdot \pmb {\omega }$, that is, $\Gamma _{\pmb {\omega }^\bullet } = \Gamma _{\pmb {\omega }}$. Moreover,

$$\begin{aligned} | \xi _\nu |^\nu (\xi _\nu \cdot \pmb {\omega } - \beta _\nu ) = | \xi _\nu |^\nu (\xi _\nu \cdot \pmb {\omega }^\bullet - (\beta _\nu + \xi _\nu \cdot \textrm{d}\pmb {\gamma })) \end{aligned}$$

and since each $\beta _\nu + \xi _\nu \cdot \textrm{d}\pmb {\gamma }$ is integral, it follows that $\pmb {\omega }^\bullet$ is strongly simultaneously approximable whenever $\pmb {\omega }$ is.

As a finitely generated Abelian group, $H_1(M; \mathbb {Z})$ admits a primary decomposition $\mathbb {Z}^d \oplus \mathbb {Z}_{p_1} \oplus \ldots \oplus \mathbb {Z}_{p_t}$. We consider $\sigma _1, \ldots , \sigma _d$ smooth 1-cycles whose homological classes form a basis of its free part $\mathbb {Z}^d$, and the map $I: H^1_{\textrm{dR}}(M; \mathbb {R}) \rightarrow \mathbb {R}^d$ given by

$$\begin{aligned} I([\alpha ]) \doteq \frac{1}{2\pi } \left( \int _{\sigma _1} \alpha , \ldots , \int _{\sigma _d} \alpha \right) . \end{aligned}$$

Since $\int _\sigma \alpha = 0$ if $\sigma$ belongs to some $\mathbb {Z}_{p_i}$, it follows that a closed 1-form $\alpha$ is integral (resp. rational) if and only if $I([\alpha ]) \in \mathbb {Z}^d$ (resp. $I[\alpha ] \in \mathbb {Q}^d$). By de Rham Theorem, we conclude that I is a linear isomorphism and that the classes of $\sigma _1, \ldots , \sigma _d$, regarded as elements of $H_1(M; \mathbb {R}) \cong H^1_{\textrm{dR}}(M; \mathbb {R})^*$, form an $\mathbb {R}$-basis for the latter vector space. We associate to $\pmb {\omega }$ the following family of vectors:

$$\begin{aligned} v_\ell \doteq \frac{1}{2 \pi } \left( \int _{\sigma _\ell } \omega _1, \ldots , \int _{\sigma _\ell } \omega _m \right) \in \mathbb {R}^m, \quad \ell \in \{1, \ldots , d \}. \end{aligned}$$

(5.13)

These, as well, depend only on $[\omega _1], \ldots , [\omega _m] \in H^1_{\textrm{dR}} (M)$ by Stokes Theorem.

Proposition 5.9

The collection $\pmb {\omega }$ is strongly simultaneously approximable if and only if there are sequences $\{ \eta _\nu \}_{\nu \in \mathbb {N}} \subset \mathbb {Z}^d$ and $\{ \xi _\nu \}_{\nu \in \mathbb {N}} \subset \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$ such that $|\xi _\nu | \rightarrow \infty$ and

$$\begin{aligned} \{ | \xi _\nu |^\nu (\xi _\nu \cdot v_\ell - \eta _{\nu \ell }) \}_{\nu \in \mathbb {N}} \quad \text {is bounded in } \mathbb {R}\text { for every } \ell \in \{ 1, \ldots , d \}. \end{aligned}$$

(5.14)

Proof

Let $Z^1 \subset \mathscr {C}^\infty (M; \Lambda ^1)$ be the space of real, closed 1-forms, endowed with the subspace topology, and define $J: Z^1 \rightarrow \mathbb {R}^d$ by $J(\alpha ) \doteq I([\alpha ])$. It is continuous—as the composition of I (a linear map between finite dimensional spaces) and the projection $Z^1 \rightarrow H^1_{\textrm{dR}}(M; \mathbb {R})$—hence maps bounded sets to bounded sets. If $\pmb {\omega }$ is strongly simultaneously approximable, there exist a sequence of closed integral 1-forms $\{ \beta _\nu \}_{\nu \in \mathbb {N}} \subset \mathscr {C}^\infty (M; \Lambda ^1)$ and $\{ \xi _\nu \}_{\nu \in \mathbb {N}} \subset \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$ such that $|\xi _\nu | \rightarrow \infty$ such that (5.12) holds. Letting $\eta _\nu \doteq I([\beta _\nu ]) \in \mathbb {Z}^d$, we have, for each $\ell \in \{1, \ldots , d\}$, that

$$\begin{aligned} J ( | \xi _\nu |^\nu (\xi _\nu \cdot \pmb {\omega } - \beta _\nu ) )_\ell = |\xi _\nu |^\nu ( J(\xi _\nu \cdot \pmb {\omega }) - J(\beta _\nu ) )_\ell = |\xi _\nu |^\nu (\xi \cdot v_\ell - \eta _{\nu \ell })\qquad \end{aligned}$$

(5.15)

must be the $\ell$-th coordinate of a bounded sequence in $\mathbb {R}^d$, yielding our first claim.

For the converse, we start with a digression. Let $\ker \Delta \subset Z^1$ denote the space of harmonic 1-forms w.r.t. some Riemannian metric, which is well-known to be finite dimensional and hence has a well-defined norm topology; we have more, the map $\alpha \in \ker \Delta \mapsto [\alpha ] \in H^1_{\textrm{dR}}(M; \mathbb {R})$ is a linear isomorphism. Moreover, $\ker \Delta$ inherits a Fréchet topology from $\mathscr {C}^\infty (M; \Lambda ^1)$, which matches the former one, by [11, Theorem 9.1]. Therefore the restriction $J: \ker \Delta \rightarrow \mathbb {R}^d$ is a linear isomorphism, hence a topological one by finiteness; in particular, an $S \subset \ker \Delta$ is bounded if and only if $J(S) \subset \mathbb {R}^d$ is bounded. We further pick $\vartheta _1, \ldots , \vartheta _d$ a basis of $\ker \Delta$ dual to $[\sigma _1], \ldots , [\sigma _d] \in H_1(M; \mathbb {R})$, in the sense that

$$\begin{aligned} \frac{1}{2 \pi } \int _{\sigma _\ell } \vartheta _{\ell '} = \delta _{\ell \ell '}, \quad \forall \ell , \ell ' \in \{1, \ldots , d\}. \end{aligned}$$

Such a choice makes $\vartheta _1, \ldots , \vartheta _d$ integral.

Now, suppose there are sequences $\{ \eta _\nu \}_{\nu \in \mathbb {N}} \subset \mathbb {Z}^d$ and $\{ \xi _\nu \}_{\nu \in \mathbb {N}} \subset \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$ such that $|\xi _\nu | \rightarrow \infty$ and (5.14) holds. Then

$$\begin{aligned} \beta _\nu \doteq \sum _{\ell = 1}^d \eta _{\nu \ell } \vartheta _\ell \in \ker \Delta \end{aligned}$$

is integral and $J(\beta _\nu ) = \eta _\nu$ for every $\nu \in \mathbb {N}$. Since (5.15) holds once more for each $\ell \in \{1, \ldots , d\}$, we have from (5.14) that $\{ J (| \xi _\nu |^\nu (\xi _\nu \cdot \pmb {\omega } - \beta _\nu ) ) \}_{\nu \in \mathbb {N}}$ must be a bounded sequence in $\mathbb {R}^d$: the conclusion follows from our digression, since we may assume w.l.o.g. $\omega _1, \ldots , \omega _m \in \ker \Delta$. $\square$

Corollary 5.10

The collection $\pmb {\omega }$ is weakly non-simultaneously approximable if and only if there exist $C, \rho > 0$ such that (see (5.13))

$$\begin{aligned} \max _{1 \le \ell \le d} | \xi \cdot v_\ell - \eta _\ell | \ge C |\xi |^{-\rho }, \quad \forall \eta \in \mathbb {Z}^d, \ \forall \xi \in \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}. \end{aligned}$$

(5.16)

Remark 5.11

• (1) Notice that

  $$\begin{aligned} \Gamma _{\pmb {\omega }} = \{ \xi \in \mathbb {Z}^m \ ; \ \xi \cdot v_\ell \in \mathbb {Z}, \ \forall \ell \in \{1, \ldots , d\} \}; \end{aligned}$$

• (2) The inequality (5.16) is equivalent to condition $\mathrm {(DC)}$ in [6, Section 2] for the $d \times m$ matrix whose rows are $v_1, \ldots , v_d$. When $\Gamma _{\pmb {\omega }} = \{0\}$, it recovers the standard notion of non-simultaneous approximability for collections of vectors in [7, Definition 1.1] (see further connections with previous conditions in the literature there);

• (3) If $v_1, \ldots , v_d \in \mathbb {Q}^m$, which corresponds to case when $\omega _1, \ldots , \omega _m$ are all rational 1-forms, we pick a non-zero $\lambda \in \mathbb {Z}_+$ such that $\lambda v_\ell \in \mathbb {Z}^m$ for every $\ell \in \{1, \ldots , d\}$. Hence, given $\xi \in \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$, there must exist an $\ell _0 \in \{1, \ldots , d\}$ for which $\xi \cdot v_{\ell _0} \notin \mathbb {Z}$, thus, for any $\eta \in \mathbb {Z}^d$, we must have

  $$\begin{aligned} \xi \cdot v_{\ell _0} - \eta _{\ell _0} \ne 0&\Longrightarrow \lambda (\xi \cdot v_{\ell _0} - \eta _{\ell _0}) \in \mathbb {Z}{\setminus } 0 \\&\Longrightarrow \lambda \max _{1 \le \ell \le d} | \xi \cdot v_\ell - \eta _\ell | \ge 1, \end{aligned}$$

  that is, (5.16) holds with $C \doteq \lambda ^{-1}$ and $\rho > 0$ arbitrary.

Now, we can state our characterization of solvability in the first degree in terms of the Diophantine condition just introduced (reworded as Theorem 1.2 in the Introduction).

Theorem 5.12

Assume $\omega _1, \ldots , \omega _m$ real-valued. The map $\textrm{d}': \mathscr {C}^\infty (\Omega ) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{1})$ has closed range if and only if the collection $\pmb {\omega }$ is weakly non-simultaneously approximable.

Proof

$(\Longrightarrow )$: Suppose there exist a sequence of integral 1-forms $\{ \beta _\nu \}_{\nu \in \mathbb {N}} \subset \mathscr {C}^\infty (M; \Lambda ^1)$ and $\{ \xi _\nu \}_{\nu \in \mathbb {N}} \subset \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$ such that $|\xi _\nu | \rightarrow \infty$ and (5.12) holds. We may assume that $\xi _\nu \ne \xi _{\nu '}$ whenever $\nu \ne \nu '$. Let $\varphi _\nu \in \mathscr {C}^\infty ({\tilde{M}}; \mathbb {R})$ be such that $\textrm{d}\varphi _\nu = \pi ^* \beta _\nu$ and define

$$\begin{aligned} f(t,x) \doteq \frac{1}{(2 \pi )^m} \sum _{\nu = 1}^\infty e^{ix\xi _\nu -i \varphi _\nu \circ \pi ^{-1}(t)} (\xi _\nu \cdot \pmb {\omega } - \beta _\nu )(t), \end{aligned}$$

which, we claim, is smooth and formally solvable. As we saw in (5.2), we have that

$$\begin{aligned} M \ni t \longmapsto e^{-i \varphi _\nu \circ \pi ^{-1}(t)} \end{aligned}$$

is a well-defined smooth function since $\beta _\nu$ is integral. Therefore, we define, for every $\xi \in \mathbb {Z}^m$, a smooth 1-form on M

$$\begin{aligned} f_\xi \doteq {\left\{ \begin{array}{ll} e^{-i \varphi _\nu \circ \pi ^{-1}} (\xi _\nu \cdot \pmb {\omega } - \beta _\nu ), &{}\quad \hbox {if }\ \xi = \xi _\nu , \\ 0, &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

Note that, in order to prove that f is a well-defined smooth 1-form, it is enough to prove that every point of M belongs to a coordinate system $(U; t_1, \ldots , t_n)$, with $\alpha \in \mathbb {Z}_+^n$ and $s > 0$, there exists $C > 0$ such that

$$\begin{aligned} \sup _U \Vert \partial _t^\alpha f_{\xi _\nu } \Vert \le C (1 + |\xi _\nu |)^{-s}, \quad \forall \nu \in \mathbb {N}, \end{aligned}$$

in which the norm $\Vert \cdot \Vert$ in the left-hand side is the sum of the absolute values of the coordinate components of the 1-form w.r.t. the local frame $\textrm{d}t_1, \ldots , \textrm{d}t_n$. By hypothesis (5.12), we may assume that, for each $\gamma \in \mathbb {Z}_+^n$, there exists $B > 0$ such that

$$\begin{aligned} \sup _U \Vert \partial _t^\gamma (\xi _\nu \cdot \pmb {\omega } - \beta _\nu ) \Vert \le B |\xi _\nu |^{-\nu }, \quad \forall \nu \in \mathbb {N}. \end{aligned}$$

It remains to prove that the derivatives of $e^{-i\varphi _\nu \circ \pi ^{-1}}$ are bounded by some constant times a power of $|\xi _\nu |$. We can assume that U is small enough in order to have a diffeomorphism $\pi _U: \tilde{U} \rightarrow U$ and so

$$\begin{aligned} e^{-i\varphi _\nu \circ \pi ^{-1}(t)} = e^{-i\varphi _\nu \circ \pi ^{-1}_U(t)} \end{aligned}$$

for every $t \in U$ and so, since $\varphi _\nu$ is real, our claim follows from the identity

$$\begin{aligned} \sum _{j = 1}^n \frac{\partial (\varphi _\nu \circ \pi _U^{-1})}{\partial t_j} \textrm{d}t_j = \beta _\nu = - (\xi _\nu \cdot \pmb {\omega } - \beta _\nu ) + \xi _\nu \cdot \pmb {\omega }, \end{aligned}$$

since the right-hand of the equality above clearly is bounded by a constant times $|\xi _\nu |$, hence proving that $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{1})$. The next step is to prove that f is formally solvable. Since ${{\hat{f}}}_\xi = 0$, if $\xi \notin \{\xi _\nu \}_{\nu \in \mathbb {N}}$, it follows that ${{\hat{f}}}_\xi$ is $\textrm{d}'_\xi$-exact for such values of $\xi$; for $\xi = \xi _\nu$ we have

$$\begin{aligned} \textrm{d}'_{\xi _\nu } e^{-i \varphi _\nu \circ \pi ^{-1}}= & {} \textrm{d}e^{-i \varphi _\nu \circ \pi ^{-1}} + i e^{-i \varphi _\nu \circ \pi ^{-1}} (\xi _\nu \cdot \pmb {\omega }) \\ {}= & {} -i e^{-i \varphi _\nu \circ \pi ^{-1}} \beta _\nu + i e^{-i \varphi _\nu \circ \pi ^{-1}} (\xi _\nu \cdot \pmb {\omega }) = {\hat{f}}_{\xi _\nu }. \end{aligned}$$

It remains to prove that there is no $u \in \mathscr {C}^\infty (\Omega )$ such that $\textrm{d}' u = f$. If such an u exists, then $\textrm{d}'_{\xi } {\hat{u}}_\xi = {{\hat{f}}}_{\xi }$ for every $\xi \in \mathbb {Z}^m$, in particular ${{\hat{u}}}_{\xi _\nu } - e^{-i\varphi _\nu \circ \pi ^{-1}} \in \ker \textrm{d}'_{\xi _\nu }$ for every $\nu \in \mathbb {N}$. But since $\xi _\nu \notin \Gamma _{\pmb {\omega }}$, we have by [4, Lemma 2.1] that $\ker \textrm{d}'_{\xi _\nu } = \{0\}$, hence ${{\hat{u}}}_{\xi _\nu } = e^{-i\varphi _\nu \circ \pi ^{-1}}$ for every $\nu \in \mathbb {N}$ and then

$$\begin{aligned} \Vert {{\hat{u}}}_{\xi _\nu } \Vert _{L^2(M)}^2 = \int _M \textrm{d}V_M, \quad \nu \in \mathbb {N}, \end{aligned}$$

does not decrease fast, contradicting the smoothness of u.

$(\Longleftarrow )$: Suppose that $f \in \mathscr {C}^\infty (\Omega ;\underline{\Lambda }^{1})$ is formally solvable, i.e., for every $\xi \in \mathbb {Z}^m$, there exists $u_\xi \in \mathscr {C}^\infty (M)$ such that $\textrm{d}'_\xi u_\xi = {{\hat{f}}}_\xi$. Thanks to Theorem 5.2, we only need to prove that there exists $u \in \mathscr {C}^\infty (\Omega )$ such that $\textrm{d}' u = f_{\mathbb {Z}^m{\setminus } \Gamma _{\pmb {\omega }}}$. The natural candidate is

$$\begin{aligned} u \doteq \frac{1}{(2\pi )^m} \sum _{\xi \in \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}} e^{ix \xi } u_\xi , \end{aligned}$$

and it suffices to prove that the series above converges in $\mathscr {C}^\infty (\Omega )$. This claim is local, so we must verify that every point of M belongs to a coordinate ball U enjoying the following property: for each $\alpha \in \mathbb {Z}_+^n$ and $s > 0$, there exists $C > 0$ such that

$$\begin{aligned} | \partial _t^\alpha u_\xi (t) | \le C(1 + |\xi |)^{-s}, \quad \forall \xi \in \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}, \ \forall t \in U. \end{aligned}$$

(5.17)

We write

$$\begin{aligned} \omega _k = \sum _{j = 1}^n \omega _{kj} \textrm{d}t_j, \quad \omega _{kj} \in \mathscr {C}^\infty (U), \end{aligned}$$

on U for $k \in \{1,\ldots ,m\}$, and

$$\begin{aligned} f = \sum _{j = 1}^n f_j \ \textrm{d}t_j, \quad f_j \in \mathscr {C}^\infty (U \times \mathbb {T}^m), \end{aligned}$$

on $U \times \mathbb {T}^m$. We may assume $\omega _{kj}$ is bounded in the $\sup$ norm in U.

Lemma 5.13

Suppose that given $s > 0$ there exists $C > 0$ such that

$$\begin{aligned} | u_\xi (t) | \le C(1 + |\xi |)^{-s}, \quad \forall \xi \in \mathbb {Z}^m{\setminus } \Gamma _{\pmb {\omega }}, \quad \forall t \in U. \end{aligned}$$

(5.18)

Then (5.17) holds true.

Proof

We proceed by induction on $|\alpha |$; the base $|\alpha | = 0$ is (5.18). For the general case, the equality $\textrm{d}'_\xi u_\xi = {{\hat{f}}}_\xi$ can be written in U as

$$\begin{aligned} \frac{\partial u_\xi }{\partial t_j} + i \sum _{k = 1}^m \xi _k \omega _{kj} u_\xi = \mathcal {F}_\xi (f_j), \quad j \in \{1, \ldots , n\}. \end{aligned}$$

If $\alpha = (\alpha _1, \ldots , \alpha _n)$, $\alpha _j > 0$ and $\beta \doteq \alpha - e_j$, in which $e_j$ is the j-th vector of the canonical basis of $\mathbb {R}^n$, then

$$\begin{aligned} \partial _t^\alpha u_\xi = \partial _t^\beta \partial _{t_j} u_\xi = \partial _t^\beta \left( \mathcal {F}_\xi (f_j) - i \sum _{k = 1}^m \xi _k \omega _{kj} u_\xi \right) , \end{aligned}$$

and we use that f is smooth and $s > 0$ in (5.17) is arbitrary. $\square$

We fix $t_0 \in M$ and prove that (5.18) holds in a coordinate ball U centered at $t_0$. As in Sect. 5.1, we assume that $\pi _U: {\tilde{U}} \rightarrow U$ is a diffeomorphism and consider the functions $\pmb {\psi }_\xi \in \mathscr {C}^\infty (\tilde{M}; \mathbb {R})$. Then on U we have, for all $\xi \in \mathbb {Z}^m$,

$$\begin{aligned} \textrm{d}(e^{i \pmb {\psi }_\xi \circ \pi _U^{-1}} u_\xi ) = e^{i \pmb {\psi }_\xi \circ \pi _U^{-1}} \textrm{d}'_\xi u_\xi = e^{i \pmb {\psi }_\xi \circ \pi _U^{-1}} {\hat{f}}_\xi \end{aligned}$$

which, integrating along any curve $\gamma$ in U connecting $t_0$ to t yields, by Stokes Theorem,

$$\begin{aligned} u_\xi (t) = e^{i ( \pmb {\psi }_\xi \circ \pi _U^{-1}(t_0) - \pmb {\psi }_\xi \circ \pi _U^{-1}(t) )} u_\xi (t_0) + e^{-i \pmb {\psi }_\xi \circ \pi _U^{-1}(t)} \int _{\gamma } e^{i \pmb {\psi }_\xi \circ \pi _U^{-1}} {{\hat{f}}}_\xi . \end{aligned}$$

Since f is smooth and $\pmb {\psi }_\xi$ is real-valued, the last equality shows that (5.18) holds true provided that we prove that, given $s > 0$, there exists $B > 0$ such that

$$\begin{aligned} | u_\xi (t_0) | \le B (1 + |\xi |)^{-s}, \quad \forall \xi \in \mathbb {Z}^m{\setminus } \Gamma _{\pmb {\omega }}, \end{aligned}$$

which we do next.

By the Hurewicz’s Theorem, we may assume that $\sigma _1, \ldots , \sigma _d$ all have base point $t_0$. We lift them to curves $\tilde{\sigma }_\ell : [0,1] \rightarrow {\tilde{M}}$ whose endpoints we denote by

$$\begin{aligned} Q_0 \doteq {\tilde{\sigma }}_\ell (0), \quad Q_\ell \doteq \tilde{\sigma }_\ell (1), \quad \ell \in \{1, \ldots , d\}. \end{aligned}$$

We integrate along ${\tilde{\sigma }}_\ell$ both sides of the equality

$$\begin{aligned} \textrm{d}(e^{i \pmb {\psi }_\xi } \pi ^* u_\xi ) = e^{i \pmb {\psi }_\xi } \pi ^* {{\hat{f}}}_\xi \end{aligned}$$

and use Stokes Theorem again to conclude that, since $\pi (Q_\ell ) = \pi (Q_0) = t_0$,

$$\begin{aligned} \left( e^{i (\pmb {\psi }_\xi (Q_\ell ) - \pmb {\psi }_\xi (Q_0))} - 1\right) u_\xi (t_0) = e^{i \pmb {\psi }_\xi (Q_0)} \int _{{\tilde{\sigma }}_\ell } e^{i \pmb {\psi }_\xi } \pi ^* {{\hat{f}}}_\xi . \end{aligned}$$

Moreover,

$$\begin{aligned} \pmb {\psi }_\xi (Q_\ell ) - \pmb {\psi }_\xi (Q_0) = \int _{\partial {\tilde{\sigma }}_\ell } \pmb {\psi }_\xi = \int _{{\tilde{\sigma }}_\ell } \textrm{d}\pmb {\psi }_\xi = \int _{{\tilde{\sigma }}_\ell } \pi ^* (\xi \cdot \pmb {\omega }) = \int _{\sigma _\ell } \xi \cdot \pmb {\omega }, \end{aligned}$$

(5.19)

so, for $\xi \in \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$, there exists $\ell$ such that $\pmb {\psi }_\xi (Q_\ell ) - \pmb {\psi }_\xi (Q_0) \notin 2 \pi \mathbb {Z}$, hence

$$\begin{aligned} u_\xi (t_0) = \frac{e^{i \pmb {\psi }_\xi (Q_0)}}{\left( e^{i (\pmb {\psi }_\xi (Q_\ell ) - \pmb {\psi }_\xi (Q_0))} - 1 \right) } \int _{{\tilde{\sigma }}_\ell } e^{i \pmb {\psi }_\xi } \pi ^* {{\hat{f}}}_\xi \end{aligned}$$

for any such $\ell$. Using again that f is smooth and $\pmb {\psi }_\xi$ is real-valued, it suffices to prove that there exist $c, \rho > 0$ such that

$$\begin{aligned} \max _{1 \le \ell \le d} \left| e^{i (\pmb {\psi }_\xi (Q_\ell ) - \pmb {\psi }_\xi (Q_0))} - 1 \right| \ge c |\xi |^{-\rho }, \quad \forall \xi \in \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}. \end{aligned}$$

(5.20)

To conclude, we use the following technical lemma, whose proof we omit.

Lemma 5.14

There exists $\epsilon > 0$ such that, for each $\xi \in \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$, at least one of the following conditions holds:

• (1) For every $\ell \in \{1, \ldots , d\}$ there exists $p_\ell \in \mathbb {Z}$ such that

  $$\begin{aligned} \left| e^{i ( \pmb {\psi }_\xi (Q_0) - \pmb {\psi }_\xi (Q_\ell ) )} - 1 \right| \ge \frac{1}{2} \left| \pmb {\psi }_\xi (Q_0) - \pmb {\psi }_\xi (Q_\ell ) - 2 \pi p_\ell \right| ; \end{aligned}$$

• (2) There exists $\ell \in \{1, \ldots , d\}$ such that

  $$\begin{aligned} \left| e^{i ( \pmb {\psi }_\xi (Q_0) - \pmb {\psi }_\xi (Q_\ell ) )} - 1 \right| \ge \epsilon . \end{aligned}$$

By Lemma 5.14, we may assume that, for every $\xi \in \mathbb {Z}^m {\setminus } \Gamma _{\pmb {\omega }}$, there are $p_1, \ldots , p_d \in \mathbb {Z}$ such that

$$\begin{aligned} \max _{1 \le \ell \le d} \left| e^{i ( \pmb {\psi }_\xi (Q_\ell ) - \pmb {\psi }_\xi (Q_0))} - 1 \right| \ge \frac{1}{2} \max _{1 \le \ell \le d} \left| \pmb {\psi }_\xi (Q_\ell ) - \pmb {\psi }_\xi (Q_0) - 2\pi p_\ell \right| . \end{aligned}$$

It follows from (5.19) and (5.13) that

$$\begin{aligned} \max _{1 \le \ell \le d} \left| e^{i ( \pmb {\psi }_\xi (Q_\ell ) - \pmb {\psi }_\xi (Q_0))} - 1 \right| \ge \pi \max _{1 \le \ell \le d} | \xi \cdot v_\ell - p_\ell |. \end{aligned}$$

Since we are assuming that $\pmb {\omega }$ is weakly non-simultaneously approximable, we can use Corollary 5.10 to conclude that (5.20) holds. $\square$

6 Formal Fourier analysis of cohomology spaces

Allowing again $\omega _1, \ldots , \omega _m$ to be complex-valued, we add a third characterization of solvability to our list in Corollary 4.8.

Proposition 6.1

The following are equivalent:

• (1) For every formally solvable $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ there exists $u \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1})$ such that $\textrm{d}' u = f$.

• (2) The map

  $$\begin{aligned} H^{q}_{\textrm{d}'}(\mathscr {C}^\infty (\Omega )) \longrightarrow \prod _{\xi \in \mathbb {Z}^m} H^q_{\xi }(\mathscr {C}^\infty (M)) \end{aligned}$$

  (6.1)

  given by $[f] \mapsto ([{\hat{f}}_\xi ])_{\xi \in \mathbb {Z}^m}$ is injective.

Proof

First, we prove that solvability implies the injectivity of (6.1). Let $[f] \in H^{q}_{\textrm{d}'}(\mathscr {C}^\infty (\Omega ))$ be such that $[{\hat{f}}_\xi ] = 0$ in $H^q_{\xi }(\mathscr {C}^\infty (M))$ for every $\xi \in \mathbb {Z}^m$, i.e., ${\hat{f}}_\xi$ is $\textrm{d}'_\xi$-exact for each $\xi \in \mathbb {Z}^m$, i.e., f is formally solvable. By the solvability, it follows that $[f] = 0$ in $H^{q}_{\textrm{d}'}(\mathscr {C}^\infty (\Omega ))$, proving the injectivity.

Conversely, a formally solvable $f \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ is $\textrm{d}'$-closed, hence it determines a class $[f] \in H^{q}_{\textrm{d}'}(\mathscr {C}^\infty (\Omega ))$. Formal solvability of f ensures that $[{\hat{f}}_\xi ] = 0$ in $H^q_{\xi }(\mathscr {C}^\infty (M))$ for every $\xi \in \mathbb {Z}^m$. Thus, if (6.1) is injective, we have $[f] = 0$ in $H^{q}_{\textrm{d}'}(\mathscr {C}^\infty (\Omega ))$, which is precisely what we wanted to prove. $\square$

Next, we study the map induced by $\mathcal {E}_\xi$ in cohomology. Unlike in the previous proposition for the map induced by $\mathcal {F}_\xi$, its injectivity always holds true and requires no extra hypotheses.

Proposition 6.2

For each $\xi \in \mathbb {Z}^m$, $\mathcal {E}_\xi$ induces an injection $H_\xi ^q (\mathscr {C}^\infty (M)) \hookrightarrow H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$. Their direct sum induces the following injection:

$$\begin{aligned} \bigoplus _{\xi \in \mathbb {Z}^m} H_\xi ^q (\mathscr {C}^\infty (M)) \longrightarrow H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega )). \end{aligned}$$

(6.2)

Proof

Let $[f] \in H_\xi ^q (\mathscr {C}^\infty (M))$. Then, by Corollary 2.9,

$$\begin{aligned} \textrm{d}' \mathcal {E}_\xi f = \mathcal {E}_\xi \textrm{d}'_\xi f = 0 \end{aligned}$$

and the class of $\mathcal {E}_\xi f$ in $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$ is well-defined. Indeed, if $[f^\bullet ] = [f]$ in $H_\xi ^q (\mathscr {C}^\infty (M))$, then there exists $v \in \mathscr {C}^\infty (M; \Lambda ^{q - 1})$ such that $\textrm{d}'_\xi v = f - f^\bullet$, hence

$$\begin{aligned} \mathcal {E}_\xi f - \mathcal {E}_\xi f^\bullet = \mathcal {E}_\xi \textrm{d}'_\xi v = \textrm{d}' \mathcal {E}_\xi v. \end{aligned}$$

Thus $[\mathcal {E}_\xi f^\bullet ] = [\mathcal {E}_\xi f]$ in $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$. Moreover, if $[\mathcal {E}_\xi f] = 0$ in $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$, then there exists $u \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1})$ such that $\textrm{d}' u = \mathcal {E}_\xi f$, hence, by Lemma 2.6, we have

$$\begin{aligned} f = \mathcal {F}_\xi \mathcal {E}_\xi f = \mathcal {F}_\xi \textrm{d}' u = \textrm{d}'_\xi {\hat{u}}_\xi \end{aligned}$$

hence the assignment $[f] \mapsto [\mathcal {E}_\xi f]$ is injective. More generally, if

$$\begin{aligned} \sum _{\xi \in \mathbb {Z}^m} [f_\xi ] \in \bigoplus _{\xi \in \mathbb {Z}^m} H_\xi ^q (\mathscr {C}^\infty (M)) \end{aligned}$$

is such that

$$\begin{aligned} \sum _{\xi \in \mathbb {Z}^m} \mathcal {E}_\xi f_\xi \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q}) \quad \text {is } \textrm{d}' \text {-exact} \end{aligned}$$

(the smoothness is possible by taking 0 as representative whenever $[f_\xi ] = 0$), then for some $u \in \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1})$, we have

$$\begin{aligned} \textrm{d}' u = \sum _{\xi \in \mathbb {Z}^m} \mathcal {E}_\xi f_\xi = \frac{1}{(2 \pi )^m} \sum _{\xi \in \mathbb {Z}^m} e^{ix \xi } \wedge f_\xi \Longrightarrow \textrm{d}'_\xi {\hat{u}}_\xi = f_\xi , \quad \forall \xi \in \mathbb {Z}^m, \end{aligned}$$

that is, $[f_\xi ] = 0$ in $H_\xi ^q (\mathscr {C}^\infty (M))$ for every $\xi \in \mathbb {Z}^m$. $\square$

7 General finiteness theorems

Assume $\omega _1, \ldots , \omega _m$ complex-valued. The next result is the heart of our forthcoming analysis.

Theorem 7.1

Given $q \in \{0, \ldots , n\}$, the following are equivalent:

• (1) $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$ is finite dimensional;

• (2) $\textrm{d}': \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1}) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ has closed range and there exists a finite set $F \subset \mathbb {Z}^m$ such that

  $$\begin{aligned} H_\xi ^q (\mathscr {C}^\infty (M)) = \{0\}, \quad \forall \xi \in \mathbb {Z}^m {\setminus } F. \end{aligned}$$

In this case:

$$\begin{aligned} H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega )) \cong \bigoplus _{\xi \in F} H_\xi ^q (\mathscr {C}^\infty (M)). \end{aligned}$$

Proof

It is a standard argument in Functional Analysis that, if $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$ is finite dimensional, then the denominator in (1.4) is a closed subspace of $\mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$. Moreover, since the map (6.2) is injective, finiteness of $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$ entails that only finitely many terms in that direct sum are non-zero. Conversely, if $\textrm{d}': \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1}) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ has closed range, then the map (6.1) is injective (by Corollary 4.8 and Proposition 6.1), i.e. $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$ injects into

$$\begin{aligned} \prod _{\xi \in F} H^q_{\xi }(\mathscr {C}^\infty (M)) \end{aligned}$$

which is finite dimensional since so is every factor and F is finite. It follows that $\dim H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega )) < \infty$, and we have a sequence of injections (6.2)-(6.1) between finite dimensional spaces

$$\begin{aligned} \bigoplus _{\xi \in F} H^q_{\xi }(\mathscr {C}^\infty (M)) \longrightarrow H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega )) \longrightarrow \prod _{\xi \in F} H^q_{\xi }(\mathscr {C}^\infty (M)) \end{aligned}$$

with isomorphic endpoints. The conclusion follows. $\square$

Below we will use that, by (3.4) and the definition of $\Gamma _{\pmb {\omega }}$ (4.3):

$$\begin{aligned} \xi \in \Gamma _{\pmb {\omega }} \Longrightarrow \mathscr {S}_{\xi \cdot \pmb {\omega }} \cong \mathscr {S}_{0} \Longrightarrow H^{q}_\xi (\mathscr {C}^\infty (M)) \cong H^q_{\textrm{dR}}(M), \quad \forall q \in \{0, \ldots , n\}. \end{aligned}$$

(7.1)

Proposition 7.2

Assume $\Gamma _{\pmb {\omega }} \ne \{0 \}$. Given $q \in \{0, \ldots , n\}$, if $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$ is finite dimensional, then $H^q_{\xi }(\mathscr {C}^\infty (M)) = \{0\}$ for every $\xi \in \mathbb {Z}^m$—in particular, $H^q_{\textrm{dR}}(M) = \{0\}$—hence $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega )) = \{0\}$.

Proof

Pick a non-zero $\eta \in \Gamma _{\pmb {\omega }}$ and a non-vanishing function $f \in \mathscr {S}_\eta (M)$. Given $u \in \mathscr {C}^\infty (M; \Lambda ^q)$ and $\xi \in \mathbb {Z}^m$, we have that

$$\begin{aligned} \textrm{d}(fu) + i ((\xi + \eta ) \cdot \pmb {\omega }) \wedge (fu)= & {} \textrm{d}f \wedge u + f \textrm{d}u + i f (\xi \cdot \pmb {\omega } + \eta \cdot \pmb {\omega }) \wedge u)\\ {}= & {} f (\textrm{d}u + i (\xi \cdot \pmb {\omega }) \wedge u) \end{aligned}$$

hence multiplication by f maps $H^q_\xi (\mathscr {C}^\infty (M))$ to $H^q_{\xi + \eta }(\mathscr {C}^\infty (M))$; since multiplication by 1/f reverses this job we have $H^q_\xi (\mathscr {C}^\infty (M)) \cong H^q_{\xi + \eta }(\mathscr {C}^\infty (M))$ for every $\xi \in \mathbb {Z}^m$. Thus, if some $H^q_\xi (\mathscr {C}^\infty (M))$ is non-zero, then so are infinitely many of them, and the left-hand side of (6.2) must contain infinitely many copies of it: injectivity of (6.2) would then lead us to a contradiction, hence proving our first claim. Now, it follows from Theorem 7.1 (with $F = \emptyset$) that $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$ vanishes. $\square$

Corollary 7.3

If $0< \dim H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega )) < \infty$ for some q, then $\Gamma _{\pmb {\omega }} = \{0\}$.

Interesting special cases of the results above are obtained for $q \in \{0, n\}$, for then $H^q_{\textrm{dR}}(M)$ is one-dimensional (compare with [4, Lemma 2.2]).

Proposition 7.4

Suppose that:

• (1) $\Gamma _{\pmb {\omega }} = \mathbb {Z}^m$;

• (2) $H^q_{\textrm{dR}}(M) = \{0\}$; and

• (3) $\textrm{d}': \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q - 1}) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{q})$ has closed range.

Then $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega )) = \{0\}$.

Proof

The first hypothesis ensures that all the factors in the direct product in (6.1) are copies of $H^q_{\textrm{dR}}(M)$, which is zero by the second hypothesis; the third one implies that $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$ embeds there, which is therefore also zero. $\square$

Theorem 7.5

If (5.11) holds, then

$$\begin{aligned} H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))\text { is finite dimensional } \Longleftrightarrow H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega )) \cong H^q_{\textrm{dR}}(M). \end{aligned}$$

(7.2)

Proof

We have by (5.11) and (7.1):

$$\begin{aligned} \bigoplus _{\xi \in \mathbb {Z}^m} H_\xi ^q(\mathscr {C}^\infty (M)) = \bigoplus _{\xi \in \Gamma _{\pmb {\omega }}} H_\xi ^q(\mathscr {C}^\infty (M)) \cong \bigoplus _{\xi \in \Gamma _{\pmb {\omega }}} H_{\textrm{dR}}^q(M) \end{aligned}$$

which is isomorphic with $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$ by Theorem 7.1 provided the latter space is finite dimensional. Two possibilities arise: either $H_{\textrm{dR}}^q(M)$ is zero (in which case $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$ vanishes too) or not; in the latter case, there must be at most finitely many indices in the last direct sum (by finite dimensionality) i.e. $\Gamma _{\pmb {\omega }}$ is a finite set. Since this is a subgroup of $\mathbb {Z}^m$, we have that $\Gamma _{\pmb {\omega }} = \{0\}$ i.e. there is a single term in that direct sum, namely $H^q_0(\mathscr {C}^\infty (M)) = H_{\textrm{dR}}^q(M)$. In both cases we get (7.2). $\square$

7.1 Calculations on surfaces

In the case $\dim M = 2$, we know that

$$\begin{aligned} \dim H_{\textrm{dR}}^q(M) = {\left\{ \begin{array}{ll} 1, &{}\hbox { if}\ q = 0, 2, \\ 2{\textsf{g}}, &{}\hbox { if}\ q = 1, \end{array}\right. } \end{aligned}$$

in which ${\textsf{g}}$ is the genus of M. It follows from the Atiyah-Singer Index Theorem that the index of the elliptic complex $\textrm{d}'_\xi$ on M, that is,

$$\begin{aligned} \sum _{q = 0}^2 (-1)^q \dim H_{\xi }^q(\mathscr {C}^\infty (M)), \end{aligned}$$

(7.3)

depends only on the principal symbol of $\textrm{d}'_\xi$, which is the same as that of the exterior derivative. In particular, (7.3) does not depend on $\xi$, hence equals

$$\begin{aligned} \sum _{q = 0}^2 (-1)^q \dim H_{\xi }^q(\mathscr {C}^\infty (M)) = \sum _{q = 0}^2 (-1)^q \dim H_{\textrm{dR}}^q(M) = \chi (M) = 2 - 2 {\textsf{g}}, \end{aligned}$$

the Euler characteristic of M. We already knew this for $\xi \in \Gamma _{\pmb {\omega }}$ by (7.1); the extra information comes for $\xi \notin \Gamma _{\pmb {\omega }}$, in which case by definition

$$\begin{aligned} H_{\xi }^0(\mathscr {C}^\infty (M)) = \mathscr {S}_{\xi \cdot \pmb {\omega }}(M) = \{0\}. \end{aligned}$$

Also, it follows from the second part of Lemma 2.8 that

$$\begin{aligned} H^2_{\xi } (\mathscr {C}^\infty (M)) \cong H^0_{-\xi } (\mathscr {D}'(M))^* \cong H^0_{-\xi } (\mathscr {C}^\infty (M))^* \end{aligned}$$

also vanishes, since $-\xi \notin \Gamma _{\pmb {\omega }}$. We conclude that for $\xi \notin \Gamma _{\pmb {\omega }}$ we have

$$\begin{aligned} \dim H_{\xi }^q(\mathscr {C}^\infty (M)) = {\left\{ \begin{array}{ll} 0, &{}\hbox { if}\ q = 0, 2, \\ 2 {\textsf{g}} - 2, &{}\hbox { if}\ q = 1. \end{array}\right. } \end{aligned}$$

Case ${\textsf{g}} = 0$. In the case M is the 2-sphere, we always have $\Gamma _{\pmb {\omega }} = \mathbb {Z}^m$, since every closed 1-form is exact by simply connectedness. Hence:

• $H_{\textrm{d}'}^{0}(\mathscr {C}^\infty (\Omega ))$ and $H_{\textrm{d}'}^{2}(\mathscr {C}^\infty (\Omega ))$ are infinite dimensional (Proposition 7.2);

• $H_{\textrm{d}'}^{1}(\mathscr {C}^\infty (\Omega ))$ is finite dimensional if and only if $\textrm{d}': \mathscr {C}^\infty (\Omega ) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^{1})$ has closed range, in which case (Theorem 7.1)

  $$\begin{aligned} H_{\textrm{d}'}^{1}(\mathscr {C}^\infty (\Omega )) \cong \bigoplus _{\xi \in \mathbb {Z}^m} H_{\xi }^1(\mathscr {C}^\infty (M)) \cong \bigoplus _{\xi \in \mathbb {Z}^m} H_{\textrm{dR}}^1(M) = \{0\}. \end{aligned}$$

Case ${\textsf{g}} = 1$. In the case M is the 2-torus, since no de Rham cohomology space vanishes, we must have that every $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$ is infinite dimensional, unless $\Gamma _{\pmb {\omega }} = \{0\}$ (Proposition 7.2). In this case we have that

$$\begin{aligned} \bigoplus _{\xi \in \mathbb {Z}^m} H_{\xi }^q(\mathscr {C}^\infty (M)) = H_{\textrm{dR}}^q(M) \oplus \bigoplus _{\xi \notin \Gamma _{\pmb {\omega }}} H_{\xi }^q(\mathscr {C}^\infty (M)) = H_{\textrm{dR}}^q(M) \end{aligned}$$

is in particular finite dimensional, hence (7.2) holds for each $q \in \{0, 1, 2\}$ (finiteness granted when $q = 0$ thanks to Theorem 7.1).

Case ${\textsf{g}} \ge 2$. As in the previous case, every $H_{\textrm{d}'}^{q}(\mathscr {C}^\infty (\Omega ))$ is infinite dimensional, except when $\Gamma _{\pmb {\omega }} = \{0\}$, in which case:

• for $q \in \{0, 2\}$, we have

  $$\begin{aligned} \bigoplus _{\xi \in \mathbb {Z}^m} H_{\xi }^q(\mathscr {C}^\infty (M)) = H_{\textrm{dR}}^q(M) \oplus \bigoplus _{\xi \notin \Gamma _{\pmb {\omega }}} H_{\xi }^q(\mathscr {C}^\infty (M)) = H_{\textrm{dR}}^q(M) \end{aligned}$$

  hence (7.2) holds, with finiteness ensured at least for $q = 0$;

• for $q = 1$, we have that

  $$\begin{aligned} \bigoplus _{\xi \in \mathbb {Z}^m} H_{\xi }^1(\mathscr {C}^\infty (M)) = H_{\textrm{dR}}^1(M) \oplus \bigoplus _{\xi \notin \Gamma _{\pmb {\omega }}} H_{\xi }^1(\mathscr {C}^\infty (M)) \end{aligned}$$

  is infinite dimensional, hence so is $H_{\textrm{d}'}^{1}(\mathscr {C}^\infty (\Omega ))$ by Proposition 6.2.

7.2 On the existence of an isomorphism under global solvability

Finally, back again to the case when $\omega _1, \ldots , \omega _m$ are real, the main result in [6]—in which $M = \mathbb {T}^n$—essentially states that, under a hypothesis that is equivalent to the global solvability in degree 1 of the operator $\textrm{d}'$, we have

$$\begin{aligned} H^{q}_{\textrm{d}'}(\mathscr {C}^\infty (\mathbb {T}^{m+n})) \cong \mathscr {C}^\infty (\mathbb {T}^r) \otimes H^q_{\textrm{dR}}(\mathbb {T}^m). \end{aligned}$$

(7.4)

This inspired us to prove our isomorphism (5.10), which considers only the cohomology of q-forms associated with the cluster $\Gamma _{\pmb {\omega }}$. We will show below that an isomorphism similar to (7.4) is not true for general M, even assuming solvability in the first degree.

Indeed, back to the case $\dim M = 2$, ${\textsf{g}} \ge 2$, let $\pmb {\omega } \doteq \{ \lambda \vartheta _1 \}$ (corank 1) in which $\vartheta _1$ is as in the proof of Proposition 5.9 ($d = 2 {\textsf{g}} > 0$ there). Therefore, if $\lambda \in \mathbb {R}{\setminus } \mathbb {Q}$, then $\Gamma _{\pmb {\omega }} = \{0\}$ thus $r = 0$, hence $\mathscr {C}^\infty (\mathbb {T}^r) \otimes H^1_{\textrm{dR}}(M) \cong H^1_{\textrm{dR}}(M)$ cannot be isomorphic with the infinite dimensional $H_{\textrm{d}'}^{1}(\mathscr {C}^\infty (\Omega ))$, even when $\textrm{d}': \mathscr {C}^\infty (\Omega ) \rightarrow \mathscr {C}^\infty (\Omega ; \underline{\Lambda }^1)$ has closed range (for instance, if $\lambda$ is a non-Liouville number, by Theorem 5.12).