Let $X$ be a complex manifold, let $E \to X$ be a smooth complex vector bundle, and let $\nabla: \Gamma(E) \to A^1(X;E)$ be a complex connection. Let the type decomposition of $E$-valued one-forms be
Let $\pi^{1,0}: A^1(X;E) \to A^{1,0}(X;E)$ and $\pi^{0,1}: A^1(X;E) \to A^{0,1}(X;E)$ be the projections onto the indicated type components, and define