The construction of a $\bar \partial$-simple covering

Abstract:

Let $M$ be a complex manifold. It is shown by simple means that an arbitrary fine open covering $\mathfrak {U} = {\{ {U_i}\} _{i \in I}}$ of $M$ exists such that for every form $\omega$ of class ${C^\infty }$ and bidegree $(p,q)$ with $\bar \partial \omega = 0$ on ${U_{{i_0}}}\bigcap \cdots \bigcap {{U_{{i_p}}}}$ there exists a form $\psi$ of class ${C^\infty }$ on ${U_{{i_0}}}\bigcap \cdots \bigcap {{U_{{i_p}}}}$ such that $\partial \psi = \omega$ provided $q \geqq 1$.