the Kähler identities
/ KAY-ler /
On any complex manifold you have several first-order operators (partial, partial-bar and their adjoints) and the Lefschetz operator L that wedges with the Kähler form, together with its adjoint Lambda. On a general complex manifold these operators interact in a complicated way. The miracle of the Kähler condition is that it forces these operators to satisfy a small set of clean commutation relations — the Kähler identities — and from those few relations the entire Hodge theory of compact Kähler manifolds unspools almost mechanically.
Precisely, on a Kähler manifold the basic identities (in one common convention) read: the commutator [Lambda, partial-bar] = -i times (the adjoint of partial), and [Lambda, partial] = i times (the adjoint of partial-bar), where Lambda is the adjoint of the Lefschetz operator L (wedging with omega). The single most consequential corollary is that the three natural Laplacians coincide up to a factor: the de Rham Laplacian, the partial-Laplacian, and the partial-bar-Laplacian satisfy Delta_d = 2 Delta_partial = 2 Delta_{partial-bar}. On a general complex (non-Kähler) manifold these three Laplacians are genuinely different operators, and the whole tidy theory collapses.
Why it matters: because the d-Laplacian equals (twice) the partial-bar-Laplacian, a form is harmonic for one exactly when it is harmonic for the other, and since the partial-bar-Laplacian preserves bidegree (p,q), the harmonic representatives — hence cohomology classes — split cleanly by type. That is the engine behind the Hodge decomposition, Hodge symmetry, and hard Lefschetz on compact Kähler manifolds. A caution: the identities are local (they hold pointwise from d omega = 0), but their dramatic cohomological consequences require compactness, where Hodge theory provides unique harmonic representatives. Conventions for signs and factors of 2, i, and 4pi vary between textbooks — always pin down the convention before quoting a formula.
The payoff in one line: on a compact Kähler manifold, Delta_d = 2 Delta_{partial-bar}. Since Delta_{partial-bar} does not mix (p,q)-types, the space of d-harmonic k-forms breaks into a direct sum over p+q=k of the (p,q)-harmonic forms. Identifying harmonic forms with cohomology classes gives the Hodge decomposition H^k = sum over p+q=k of H^{p,q} directly.
The identities force Delta_d = 2 Delta_{partial-bar}, and bidegree-preservation then splits cohomology by type.
The equality of the three Laplacians is exactly what fails on a non-Kähler complex manifold; it is the Kähler condition d omega = 0, not mere complexness, that makes Hodge theory work. Sign and factor conventions differ across texts.