Dolbeault operators and Dolbeault cohomology

Use Wirtinger derivatives with one half in the definitions: differentiation with respect to z is half of real-x differentiation minus i times real-y differentiation, and conjugate differentiation uses the plus sign. On smooth functions, partial collects the dz terms and bar-partial collects conjugate differential terms. Extend by the graded Leibniz rule. On an integrable complex manifold the different bidegrees in the square of d must vanish separately. This yields partial squared zero, bar-partial squared zero, and anticommutation of the two operators.

For each fixed holomorphic degree p, bar-partial therefore forms a complex in the antiholomorphic degree q. Closed means the next bar-partial is zero; exact means the form is the bar-partial of a global smooth form of degree one less. The cohomology quotient distinguishes these conditions. Holomorphic p-forms occupy the zero-antiholomorphic-degree kernel. In positive degree, exactness always implies closedness but the converse is a local assertion only until global obstructions have been addressed.

The local Dolbeault lemma says every smooth bar-partial-closed form with positive antiholomorphic degree is locally bar-partial-exact. Consequently the sheaves of smooth p-q forms give a fine resolution of the sheaf of holomorphic p-forms. Smooth partitions of unity make those resolution sheaves acyclic on the underlying paracompact manifold. Taking global sections computes sheaf cohomology and produces the displayed Dolbeault isomorphism. This explanation separates the local analytic input from the global topological mechanism.

More generally, let E be a holomorphic vector bundle of finite rank with its integrable bar-partial operator. Its smooth bundle-valued Dolbeault complex computes cohomology of the locally free sheaf of holomorphic sections, with the holomorphic p-form factor included. No compactness or Kähler assumption is needed for this sheaf identification. An arbitrary coherent sheaf need not itself be a vector bundle, so one cannot simply write the same bundle-valued smooth-form complex for it without additional resolutions or constructions.

On a flat complex torus, the constant conjugate one-form descends globally from the plane. It is bar-partial-closed. On the plane it has a simple primitive, the conjugate coordinate, but that coordinate is not periodic and hence is not a global function on the quotient. A global primitive on the torus would contradict integration against a holomorphic one-form: Stokes' theorem kills the integral of a global exact expression, while the constant volume pairing is nonzero. This is a concrete obstruction rather than merely a lack of an obvious formula.

Care with antisymmetric signs is equally important locally. Differentiating a coefficient creates a conjugate differential on the left. Reordering it past another conjugate differential changes the sign. Two terms can cancel for precisely that reason, while a superficially similar single term remains nonclosed. The scalar quadratic instrument checks the real-axis restriction of a simple complex potential; the full bar-partial derivative must still be calculated from both real variables, not inferred from one real plot alone.