the ∂∂-bar lemma
/ del del-bar lemma /
Suppose a differential form on a compact Kähler manifold is exact in three different senses at once — it is d-exact, and partial-exact, and partial-bar-exact. The natural question is whether you can write it down explicitly as coming from a single potential. The partial partial-bar lemma says yes, and in a very clean way: any form that is exact for any of these operators (and closed for the others) is actually equal to partial partial-bar of one global form. It is the global, complex-geometric upgrade of the Poincaré lemma, and it is the technical workhorse behind much of Kähler and Calabi-Yau theory.
Precisely, on a compact Kähler manifold (more generally on any manifold where the lemma holds), if a (p,q)-form alpha is both partial-closed and partial-bar-closed (equivalently d-closed) and is d-exact, then there exists a (p-1,q-1)-form beta with alpha = partial partial-bar beta. The various exactness notions collapse: ker(d) intersect (image of partial, partial-bar, or d) all coincide and equal image(partial partial-bar). A central application is to Kähler classes: any two Kähler forms in the same de Rham cohomology class differ by partial partial-bar of a global real function f (the relative Kähler potential), omega' = omega + i partial partial-bar f. This reduces questions about whole cohomology classes of metrics to a single scalar equation — exactly the setup for the Calabi conjecture, whose solution (Yau's theorem) becomes a Monge-Ampere PDE for that potential f.
Why it matters: the lemma is what lets you 'integrate' curvature conditions to potentials and turn geometric PDE on a Kähler manifold into scalar PDE. It also gives quick proofs that the Dolbeault and de Rham pictures match (formality of compact Kähler manifolds). The honesty point: the partial partial-bar lemma is a theorem about compact Kähler manifolds (and a few generalizations such as manifolds in class C of Fujiki), not about arbitrary compact complex manifolds. On a non-Kähler complex manifold it generally fails, and that failure is measured by the Bott-Chern and Aeppli cohomologies, which differ from Dolbeault cohomology precisely there.
Two Kähler metrics omega and omega' on a compact Kähler manifold representing the same class in H^2(M, R) must satisfy omega' - omega = i partial partial-bar f for a single real function f. So studying all Kähler metrics in a fixed class reduces to studying scalar functions f with omega + i partial partial-bar f still positive — the variable in the Calabi-Yau Monge-Ampere equation.
Same Kähler class means omega' = omega + i partial partial-bar f: metrics become one scalar function f.
The lemma holds on compact Kähler manifolds, not arbitrary compact complex ones. Its failure on non-Kähler manifolds is exactly what Bott-Chern and Aeppli cohomologies detect — do not assume it without the Kähler hypothesis.