Global properties of the differential complex associated to closed, nonsingular $1$-forms on compact manifolds

Given a closed, real, non-singular 1-form on a compact manifold $\mathrm{\Omega }$, global properties of the associated differential complex are studied. We completely characterize global solvability in the first and last levels of the complex. Furthermore, in the particular case where the $1$-form is rational, we prove global solvability for every degree and give a complete description of the cohomology spaces. Finally, a complete characterization for global hypoellipticity is obtained, building on the work of A. Meziani (Comm. PDE., 2002). In all cases, it is shown that the conditions depend exclusively on the arithmetic nature of the form’s periods.