Vanishing Theorems, Support Conditions, and Boundary Problems for \(\overline\partial\) on Weak Z(q) Domains

Let \( X \) be a complex manifold of complex dimension \( n \geq 2 \), and let \( \Omega \Subset \mathcal{X} \) be a relatively compact domain with smooth boundary that satisfies the weak \( Z(q) \)-condition. Assume \( \mathcal{F} \) is a holomorphic line bundle over \( X \), and denote by \( \mathcal{F}^{\otimes m} \) its \( m \)-th tensor power for some positive integer \( m \). Provided there exists a strongly plurisubharmonic function defined in a neighborhood of the boundary \( b\Omega \), it is possible to obtain solutions to the \( \overline{\partial} \)-equation within \( \Omega \), under support conditions, for \((p,q)\)-forms with \( q \geq 1 \) taking values in \( \mathcal{F}^{\otimes m} \). Additionally, we study the solvability of the boundary \( \overline{\partial}_b \)-problem on weak \( Z(q) \)-domains with smooth boundary in the setting of Kähler manifolds. Moreover, an extension theorem for \( \overline{\partial}_b \)-closed differential forms will be proven.