Lemma: $\omega\in\bigwedge^2(T)$ where $T$ is torus($(S^1)^m)=\frac{C^m}{\Lambda}$ for some $m\in Z_{>0}$ and $\bigwedge^n(T)$ is the smooth $n-$form on torus. Suppose $d\omega=0$ on torus. Denote $\omega^\star$ the pull back of $\omega$ to $C^m$ via $C^m\to T=(S^1)^m\cong\frac{C^m}{\Lambda}$ where $\Lambda$ is some lattice. Then $\exists\psi\in\bigwedge^1(T)$ and $\phi\in\bigwedge^1(C^n)$ s.t. $\omega^\star=d\psi+d\phi$. If $\eta_i$ are coordinates of $C^n$, then $\phi=\sum_{ij}c_{ij}\eta_id\eta_j$ for some $c_{ij}\in C$.
From the proof, it seems that $\phi$ is really $0$ mode/constant term contribution of $\omega$ but $\psi$ is the higher mode contribution.($\psi$ is higher mode contribution due to the fact that $\psi$ is really a form on torus.) If given fourier mode construction of $\omega$, then I can construction $\phi$ and $\psi$ explicitly.
$\textbf{Q:}$ I think it should be intuitive to use fourier analysis here as it is a form on torus which is exactly periodicity comes in. What is the intuitive reason that I would expect a $2-$form on torus written in terms of exterior differential of $1-$form on torus with higher mode information and constant mode information lying on $C^n$?(Why do I expect this phenomena?) This is the technical lemma used to prove that every multiply periodic function is a quotient of theta functions. Is this expected in harmonic analysis on forms here?
$\textbf{Q':}$ I think the result should generalize to $n$ forms on torus as well. The prescription should be similar but $\phi$ needs some modification here. Is this correct?
Ref: Analytic Theory of Abelian Varieties by Swinnerton-Dyer.