Let $X$ be a complex manifold of complex dimension $n$, and let $\pi: E \to X$ be a holomorphic vector bundle. For every pair of integers $p,q$ with $0 \leq p,q \leq n$, let $\bar\partial_E: A^{p,q}(X,E) \to A^{p,q+1}(X,E)$ denote the Dolbeault operator induced by the holomorphic structure on $E$. Then
as a map $A^{p,q}(X,E) \to A^{p,q+2}(X,E)$. Consequently, for each fixed $p$, the sequence $A^{p,\bullet}(X,E)$ with differential $\bar\partial_E$ is a cochain complex.