# ag.algebraic geometry – Global sections appearing in Dolbeault complex with values in vector bundle

Given a holomorphic vector bundle $$E$$ on a compact complex Kähler manifold $$X$$ (I am happy to assume $$X$$ projective), we can compute the sheaf cohomology $$H^ast(E)$$ of $$E$$ using the Dolbeault complex with values in $$E$$ (see e.g. Huybrechts’ book Complex Geometry, Section 2.6).

The Dolbeault complex consists of the (smooth) global sections $$Gamma(X, mathcal{A}^{0,q}_X otimes E)$$ of the (smooth) bundle $$mathcal{A}^{0,q}_X otimes E$$. Is it known whether the (infinite-dimensional) vector space $$Gamma(X, mathcal{A}^{0,q}_X otimes E)$$ is generated by tensor products $$omega otimes v$$ with $$omega$$ respectively $$v$$ smooth sections of $$mathcal{A}^{0,q}_X$$ respectively $$E$$? I believe this does not need to hold.

Note that the description of the question Global Definition of the Dolbeault Complex of a Vector Bundle seems to assume this is true, whereas the comment underneath says that the operator $$bar{partial}_E$$ only locally agrees with $$bar{partial} otimes id$$ (which I interpret as a negative answer to my question).