Stein vanishing for the Dolbeault cohomology of functions ($H_{\bar{\partial }}^{0,q}(X)=0,q>0$)

On a Stein manifold, every smooth $\bar{\partial }$-closed $(0,q)$-form with $q>0$ is globally $\bar{\partial }$-exact. The same holds componentwise for forms valued in a finite-dimensional constant vector space. Complex conjugation yields the corresponding global $\partial$ primitive for a closed $(1,0)$-form, as used in the complex potential reduction of anti-self-dual Yang-Mills.