Affine space of solutions of the Hodge Poisson equation (${\alpha }_0+{\mathcal{H}}^k(M)\cong {\alpha }_0+H_{\mathrm{dR}}^k(M)$)

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.