(p,q)-form

Summary

A $(p, q)$-form on an open subset of $\C^n$, or on a complex manifold, is a complex-valued differential form of a special type. Each of its terms contains $p$ of the differentials $dz_j$ and $q$ of the conjugate differentials $d\bar z_j$. Every complex differential form of degree $k$ splits in a unique way as a sum of $(p, q)$-forms with $p + q = k$. This splitting does not depend on the choice of holomorphic coordinates. It is the setting of the $\bar\partial$ operator and of Dolbeault cohomology.