= Affine space of solutions of the Hodge Poisson equation
{title2=$\alpha_0+\mathcal H^k(M)\cong\alpha_0+H^k_{\mathrm{dR}}(M)$}
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.
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