the (p,q)-type decomposition of forms

On a complex manifold the coordinate differentials come in two flavours: the holomorphic ones dz^1, ..., dz^n and their conjugates d(z-bar)^1, ..., d(z-bar)^n. Any differential form can be sorted according to how many of each kind it contains. A form built from p of the dz's and q of the d(z-bar)'s is said to have type (p,q), or bidegree (p,q). The (p,q)-decomposition is the bookkeeping that splits every complex-valued differential form into these pure-type pieces — the single most useful organizing principle in complex geometry.

Precisely, the complexified tangent space splits, via the almost-complex structure J, into the +i-eigenspace (holomorphic directions) and the -i-eigenspace (antiholomorphic directions). Dualizing, the complexified k-forms split as a direct sum over all p + q = k of the spaces of (p,q)-forms, written as a sum of bundles Lambda^{p,q}. A (p,q)-form looks locally like a sum of terms f times (dz^{i_1} ^ ... ^ dz^{i_p}) ^ (d(z-bar)^{j_1} ^ ... ^ d(z-bar)^{j_q}) with smooth coefficient f. The exterior derivative d then splits as d = partial + partial-bar, where partial raises p by one and partial-bar raises q by one — this splitting is exactly what makes the Dolbeault operators well defined, and it is available precisely because J is integrable (Newlander-Nirenberg); on a non-integrable almost-complex manifold d has extra components and the clean splitting fails.

This decomposition is why complex manifolds carry strictly more cohomological information than their underlying real manifolds: de Rham cohomology gets refined by bidegree into Dolbeault cohomology, and on a compact Kähler manifold the two are linked by the Hodge decomposition H^k = sum over p+q=k of H^{p,q}. A common slip: the (p,q)-splitting of forms exists on any almost-complex manifold, but the clean factorization d = partial + partial-bar (with no other components) requires integrability; without it, d also has (2,-1) and (-1,2) pieces coming from the Nijenhuis tensor.