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.

Principle

Hodge theory: on a compact Kähler manifold the Laplace operator associated to the metric yields finite-dimensional spaces of harmonic forms whose direct sum, organized by Hodge type (p,q), equals the total cohomology.

Demonstration

For a compact Riemann surface of genus g, H^1_{dR} splits as H^{1,0} ⊕ H^{0,1}; holomorphic 1-forms give the H^{1,0} summand and their complex conjugates give H^{0,1}, so the 2g-dimensional real cohomology is accounted for by g complex holomorphic forms and their conjugates.

Misapplication

Assuming the same bidegree splitting holds for arbitrary complex manifolds without the Kähler condition; on non-Kähler complex manifolds H^{p,q} spaces need not combine to reconstruct de Rham cohomology orthogonally.

Consequence

Provides numerical invariants (Hodge numbers) restricting topology and complex structure, yields Hodge symmetry and relations between Betti and Hodge numbers, and lets one compute cohomology via holomorphic data.

Reversal

Inverting the statement gives a situation where cohomology does not split by bidegree: for a noncompact or non-Kähler manifold harmonic representatives may exist but need not produce a bidegree decomposition of cohomology.

Boundary

Requires a compact Kähler manifold for the full Hodge decomposition into (p,q)-types; for general compact Riemannian manifolds there is a Hodge isomorphism between harmonic forms and de Rham cohomology but no refined (p,q) splitting.

Semantic Tension

Confused with the Hodge filtration or with mere identification of harmonic forms; the decomposition is a direct-sum splitting by bidegree, while the Hodge filtration is a descending filtration encoding related but different data.

Synthesis

Hodge decomposition is the statement that, on a compact Kähler manifold, metric harmonicity and complex bidegree structure combine to split de Rham cohomology into orthogonal (p,q) summands, producing computable Hodge numbers that reflect both topology and complex geometry.