Solvability condition for the Hodge Poisson equation ($\beta \perp {\mathcal{H}}^k(M)$)

The Poisson equation for differential forms on a closed oriented Riemannian manifold is solvable exactly when its source is $L^2$-orthogonal to every harmonic differential form of the same degree. Necessity follows from the formal adjoint identity $⟨\Delta \alpha ,h⟩=⟨\alpha ,\Delta h⟩=0$. For sufficiency, the Green operator of the Hodge Laplacian gives $\alpha =G\beta$, since $\Delta G=I-P_{\mathcal{H}}$.

When the Poisson equation for differential forms has one solution ${\alpha }_0$, all solutions are its translates by the kernel of a linear map of the Hodge Laplacian, namely the harmonic differential forms. The Hodge decomposition theorem identifies this translation vector space canonically with de Rham cohomology. There is no preferred affine origin until an extra normalization is chosen; orthogonality to harmonic forms picks the Green-operator solution.