Hodge Decomposition

An orthogonal splitting of the space of differential k‑forms on a compact oriented Riemannian manifold into exact, coexact, and harmonic parts: every form equals dα + δβ + γ with dγ = δγ = 0.

A canonical orthogonal splitting of the de Rham cohomology of a compact Kähler manifold into subspaces of differential forms of fixed complex bidegree, realized by harmonic representatives for each bidegree.

An orthogonal decomposition of the space of differential forms on a compact Riemannian manifold into three mutually orthogonal subspaces: harmonic forms (kernel of the Hodge Laplacian), exact forms (images of the exterior derivative), and coexact forms (images of the codifferential).

A decomposition of the complex cohomology of a suitable geometric object (classically a compact Kähler manifold or a smooth projective variety) into a direct sum of H^{p,q} pieces, H^n(X,C) = ⊕_{p+q=n} H^{p,q}(X), compatible with complex conjugation and relating analytic (Dolbeault), algebraic and topological invariants via harmonic representatives and the Hodge filtration.