= Stein vanishing for the Dolbeault cohomology of functions
{c}
{title2=$H^{0,q}_{\bar\partial}(X)=0,\quad q>0$}
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>.
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