Communications of the
Korean Mathematical Society CKMS

Let $X$ be a complex manifold of dimension $n$\geqslant 2 and let $\Omega$\Subset $X$ be a weakly $q$-convex domain with smooth boundary. Assume that $E$ is a holomorphic line bundle over $X$ and $E^{\otimes m}$ is the $m$-times tensor product of $E$ for positive integer $m$. If there exists a strongly plurisubharmonic function on a neighborhood of $b\Omega$, then we solve the $\overline{\partial}$-problem with support condition in $\Omega$ for forms of type $(r,s)$, $s$\geqslant $q$ with values in $E^{\otimes m}$. Moreover, the solvability of the $\overline{\partial}_{b}$-problem on boundaries of weakly $q$-convex domains with smooth boundary in K\"{a}hler manifolds are given. Furthermore, we shall establish an extension theorem for the $\overline\partial_{b}$-closed forms.

Keywords: $\overline{\partial}$ and $\overline{\partial}$-Neumann operators, pseudoconvex domains, line bundle

MSC numbers: 32F10, 32W05, 32W10, 35J20, 35J60