Let $X$ be a complex manifold, let $\pi: E \to X$ be a holomorphic vector bundle, and let $\bar{\partial}_E: C^\infty(X,E) \to \Omega^{0,1}(X,E)$ denote the Dolbeault operator determined by the holomorphic structure on $E$. For every smooth section $s \in C^\infty(X,E)$, the section $s$ is holomorphic if and only if
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\begin{align*}
\bar{\partial}_E s = 0.
\end{align*}