Vector-bundle-valued differential form (${\mathcal{A}}^k(E)$)

A smooth section of ${\bigwedge }^k(T^{\ast }X\otimes \mathbb{C})\otimes E$ is a differential form with values in the complex vector bundle $E$. A connection on a vector bundle extends to these forms by the graded Leibniz rule.

For a holomorphic vector bundle $E$, this sheaf assigns to $U$ the smooth $E$-valued differential forms of type (p, q) on $U$. Its Dolbeault operator acts coefficientwise in a holomorphic local frame and squares to zero.