Hodge decomposition of a compact Kähler manifold

Statement

Let $X$ be a compact Kähler manifold. For every $k$ , the Hodge decomposition is the canonical direct sum

H^k_{\mathrm{dR}}(X;\mathbb C) \cong \bigoplus_{p+q=k}H_{\bar\partial}^{p,q}(X)

of Dolbeault cohomology groups. Under the Hodge theorem, a de Rham class has a unique harmonic representative, and the displayed isomorphism sends it to its harmonic components of type $(p,q)$ . Complex conjugation exchanges the summands $H^{p,q}(X)$ and $H^{q,p}(X)$ . Compactness and the Kähler condition are part of the theorem.

Analytic mechanism

The Kähler identities imply

\Delta_d=2\Delta_{\partial}=2\Delta_{\bar\partial}.

Consequently the de Rham Laplacian preserves bidegree, so each $(p,q)$ -component of a harmonic form is harmonic. This is the step that turns the type decomposition of differential forms into a direct-sum decomposition of cohomology Voisin, §6.1, Theorem 6.3.

Consequences

Writing $h^{p,q}(X)=\dim_{\mathbb C}H^{p,q}(X)$ , the decomposition gives

b_k(X)=\sum_{p+q=k}h^{p,q}(X)

and conjugation gives $h^{p,q}=h^{q,p}$ . In particular, every odd Betti number of a compact Kähler manifold is even. Wedge product respects bidegree, so the decomposition is compatible with the graded cohomology ring.

Scope and near-misses

A compact complex manifold need not admit this decomposition. The Frölicher spectral sequence may have nonzero higher differentials, and even its degeneration does not by itself provide the harmonic splitting above. For noncompact Kähler manifolds, ordinary de Rham cohomology need not be represented by harmonic forms without additional analytic conditions.

References

Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002. Publisher record. Relevant: §6.1, especially Theorem 6.3.
Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Publisher record. Relevant: Chapter 3, §3.2, Hodge decomposition and its numerical consequences.