(p,q)-form

Remarks

The forms $dz_I \wedge d\bar z_J$ with $\abs{I} + \abs{J} = k$ form a basis of the complex $k$-th exterior power of the cotangent space at each point. So every complex $k$-form is, in a unique way, a sum of $(p, q)$-forms with $p + q = k$. Example: $\omega = \bar z_1 \, dz_1 \wedge d\bar z_2$ on $\C^2$ is a $(1, 1)$-form, with $\omega_{(1)(2)} = \bar z_1$ and all other coefficients zero. Non-example: $dx_1 = \tfrac12 dz_1 + \tfrac12 d\bar z_1$ is a $1$-form whose $(1, 0)$ and $(0, 1)$ parts are both nonzero, so it is not a $(p, q)$-form for any $(p, q)$. A holomorphic change of coordinates $w = F(z)$ gives $dw_k = \sum_j \frac{\partial F_k}{\partial z_j} dz_j$ with no $d\bar z_j$ terms, so the type $(p, q)$ is preserved; thus $(p, q)$-forms are defined on complex manifolds. The space of $(p, q)$-forms on $U$ is often written $\mathcal{E}^{p, q}(U)$.