Sheaf of smooth differential forms (${\mathcal{A}}^k$)

On a smooth manifold, ${\mathcal{A}}^k(U)$ is the space of smooth complex-valued degree-$k$ differential forms on $U$. Restriction of forms supplies the sheaf maps. Smooth partitions of unity make this a fine sheaf.

On a complex manifold, this sheaf consists of smooth differential forms of type (p, q). The type decomposition of the complexified tangent bundle gives ${\mathcal{A}}^k={⨁}_{p+q=k}{\mathcal{A}}^{p,q}$.